What does "free and independent states" mean in the Declaration of Independence?

Short Answer:

The words in the documents quoted can be interpreted in two different ways, and the correct interpretation is probably different from the one in the question. Also, what could be called the "Second American Revolution" happened after those documents and before the Civil War, and revolutionized the governmental structure of the USA.

Long Answer:

On June 11, 1776, the Continental Congress selected a committee to write a Declaration of Independence.

One June 12, 1776, the Continental Congress decided to appoint a committee to write a constitution for a union of the states.

So one might argue that the US leadership only intended for each former colony to be an independent nation for a day before they decided that there had to be a united government of all the former colonies.

The Declaration of Independence was made in July, 1776, while the rules for a union of the states were still being worked out. Therefore, it was more or less technically correct to described the former colonies as (desiring and striving to be) independent and sovereign states in July 1776 - subject to the outcome of the war, of course.

Note that the Declaration of Independence does not refere to the "States of America" but to the United States of America". Thus it says that the various colonies/states are in some way, manner, or form, united. It might refer to them as 13 totally separate and independent governments united by a common goal, t seek their separate independence from Great Britain. Or it might refer to them as members of a sort of a league, such as the Dutch Republic, or the decaying Hanseatic League, or the Old Swiss Confederacy, organizations composed of several member groups that were treated as more or less single states for the purpose of diplomacy.

The final draft of the articles of Confederation was completed in November, 1777, and was submitted to the colonies/states for their ratification.

The Articles of Confederation was submitted to the states for ratification in late November 1777. The first state to ratify was Virginia on December 16, 1777; 12 states had ratified the Articles by February 1779, 14 months into the process.[11] The lone holdout, Maryland, refused to go along until the landed states, especially Virginia, had indicated they were prepared to cede their claims west of the Ohio River to the Union.[12] It would be two years before the Maryland General Assembly became satisfied that the various states would follow through, and voted to ratify. During this time, Congress observed the Articles as its de facto frame of government. Maryland finally ratified the Articles on February 2, 1781. Congress was informed of Maryland's assent on March 1, and officially proclaimed the Articles of Confederation to be the law of the land.[11][13][14]

https://en.wikipedia.org/wiki/Articles_of_Confederation#Ratification1

And once the various former colonies began acting under the Article of Confederation, they were no longer totally separate, independent, an d sovereign states, not matter what the wording of the Articles said. The United States of America was now a functioning confederation or federation that was vaguely similar in various ways to the Dutch Republic or the Old Swiss Confederacy in Europe, or to the Iroquois League in America, or to the Western Confederacy that formed in 1783 to resist American expansion in the Ohio Country.

And note that the union formed by the Articles of Confederation is said to be perpetual. This implies that the sovereignty of the states is limited and they are unable to legally withdraw from that union.A

Meanwhile the war had been dragging on for five or six long years and both sides were at the end of their ropes, almost at the breaking point. On October 19, 1781, the army of Lord Cornwallis surrendered at Yorktown. Even though the British still had a powerful army in the colonies, the news broke the British will to continue fighting. The prime minister Lord North said "It's all over." when he heard about Yorktown.

Negotiations began soon, and the treaty of Paris was signed on September 3, 1783, ratified by the USA on January 14, 1784, and by Britain on April 9, 1784, and the ratified copies were exchanged on May 12, 1784.

https://en.wikipedia.org/wiki/Treaty_of_Paris_(1783)2

And all the European diplomats considered the United States of America to be a functioning government composed of other, subordinate, governments vaguely like the Dutch Republic or the Old Swiss Confederacy or the Holy Roman Empire. Foreign diplomats considered the United States of America to be one independent state that contained 13 very autonomous dependent states within it.

So the wording of the Treaty of Paris listed all the states to make sure everyone knew that each and every single one of them was included in the terms of the treaty. And maybe another reason the British agreed to suggested that wording in the treaty was to sort of insult the rather weak and feeble central government of the USA which had defeated them by implying there wasn't any central government in the USA at all and that each state was totally sovereign and independent.

But in any case, the meaning of the words in the Declaration of Independence, the Articles of Confederation, and The treaty of Paris must be less than each state being totally free, totally sovereign,and totally independent, because the leaders of the 13 colonies were already working on a plan for a form of confederation that would make each of the colonies less than totally fee, less than totally sovereign, and less than totally independent, when the Declaration of Independence was approved, so it was always their intention to make the USA a single country, although one with a lot of autonomy for the various states, vaguely similar to the Dutch Republic or the Old Swiss Confederacy.

The meaning of those words must have been that the United States of America was a group of 13 states that was totally free, sovereign, and independent of Great Britain, not thirteen separate states that were totally free, sovereign, and independent of each other.

And then something major happened after the Treaty of Paris in 1783 and before the Civil War in 1861-1865.

What could be the called the Second American Revolution happened. It is not called a revolution because there were no laws broken, and no bloodshed. But, like a number of other and later revolutions in the USA that are not usually recognized as revolutions, its results were just as revolutionary as the American Revolution with its thousands of deaths. This revolution completely changed the structure of the American government.

The Constitutional Convention met in 1787 and designed the Constitution of the United States of America, "a new and ore perfect union", which a much stronger central government.

One by one the 13 states ratified the new Constitution. Delaware ratified it on December 7, 1787. New Hampshire ratified the constitution in June 21, 1788. Since it was the ninth state to do so, and that was the agreed upon number, the ratification by New Hampshire guaranteed that the new constitution would come into effect among those states which had already ratified it and any states with later did so. Virginia ratified on June 25 and New York on July 21, 1788.

On September 1788, the Congress of the Confederation decreed that the Constitution had been ratified and set the date for the new federal government to start operations. The first presidential election was held in December 1788 to January 1789. The presidential electors met to cast their votes on February 4, 1789. The United States Congress met for the first time on March 4, 1789. George Washington began his first time as President on April 30, 1789. On November 21, 1789, North Caroline ratified the Constitution.

The Supreme Court of the USA met for the first time on February 2, 1790. Rhode Island became the last of the 13 original colonies and states to ratify the US constitution on May 29, 1790.

https://en.wikipedia.org/wiki/Timeline_of_drafting_and_ratification_of_the_United_States_Constitution3

None of the other 37 states in the USA had anything to do as a state with creating the USA.

West Virginia was created out of the state of Virginia with the agreement of the loyalist government of Virginia. Two other other states were created when separate and independent countries, The Vermont Republic, and the Republic of Texas, applied to join the USA and were admitted.

The other thirty four states of the USA were created by the Federal government out of lands owned by the federal government, lands which the federal government acquired by various methods from foreign governments such as Great Britain, The United Kingdom of Great Britain and Ireland, The French Empire, Spain, Mexico, the Russian Empire, and the Republic of Hawaii.

Those other thirty four states of the USA were not among the 13 free and independent states mentioned in the Declaration of Independence, And they did not have their "sovereignty, freedom, and independence" mentioned in the Articles of Confederation, nor were they mentioned by name as "free sovereign and independent states" in the Treaty of Paris.

So the wording in those documents has no application to the status of those thirty four states of the United States during the period of history after they were admitted as States in the United States. And since all states in the union are of equal status, the words in those documents can not apply to any of the thirteen original states after they ratified the constitution and became states in the new federal constitution of the USA. Even if those worlds should be interpreted as meaning that the states were totally free, totally sovereign, and totally independent, which seems unlikely anyway.