What are some examples of US states warring with one another?

The entire state of Vermont is territory that broke away from New York.

The area was originally controlled by New Hampshire but was transfered to New York. The local New-Hampshire-recognized owners rebelled when the New York issued new deeds covering their land. Rebel forces were called the Green Mountain Boys.

I'll try to give a more detailed analysis of the conflict between Vermont and New York than the short answer posted here earlier. It never errupted into military conflict but in 1784 Governor George Clinton of New York asked Congress to authorize military force against Vermont. They didn't, and Clinton didn't follow through on his threat to use New York militiamen to invade Vermont.

Whether Vermont "broke away" from New York depends on whether New York's claim to Vermont was valid. Certainly King Charles II in 1664 decided the west bank of the Connecticut River was to be the eastern boundary of New York, so that it included what later became Vermont.

In 1749 Governor Benning Wentworth of New Hampshire issued the first of his well over 100 "grants". Each of these established a "town" that was typically 6 miles by 6 miles, thus 36 square miles, and offered land there for sale. The first grant was the town of Bennington, which he named after himself. Numerous settlers arrived over a 15-year period starting in 1749, and they thought they lived in New Hampshire.

In 1764 King George III ruled that the turf west of the Connecticut River belonged to New York. The king's ruling followed an ex parte hearing, i.e. he heard from only one side of the dispute: the government of New York. That was one reason that who disagreed thought they might convince him to change his mind.

The colonial government of New York auctioned off land in the region that was then called the New Hampshire Grants and was not yet known as Vermont. These "land patents" from New York conflicted with grants from New Hampshire, so the many New Hampshire grantees were threatened with eviction. The Green Mountain Boys were a militia formed to resist such evictions, and local governments made it an offense punishable by flogging, forfeiture of all property, and banishment, to accept a commission as a sheriff or a justice of the peace from New York.

In 1767 the king ordered New York to recognize the claims of settlers who had grants from New Hampshire, but that was not well enforced.

In 1775 the Green Mountain Boys were made a part of the Continental Army and their leader, Ethan Allen, was made a colonel.

In January 1777 the politicians of the New Hampshire Grants assembled and published a list of grievances against both King George III and the state of New York, and declared what they called "New Connecticut" a separate state. In July they decided to call their new state Vermont.

Vermont was denied representation in the Continental Congress because of objections from New York. This happened repeatedly over a number of years, I think until about 1785.

Vermont made the offense of treason against Vermont punishable by forfeiture of all property and banishment, and treason meant supporting either George III or the claims of New York to Vermont.

In August 1781, after the Articles of Confederation came into effect, Congress said that Vermont needed only to give up its claims to turf east of the Connecticut River and west of Lake Champlain in order to be admitted to the Union. Under the Articles, each of the 13 states could cast one vote on each measure in Congress, and admission of a new state required nine votes. Throughout the 1780s, there was a lot of talk about admitting new states, including Vermont, Kentucky, Franklin, and Maine. (Franklin was of course never admitted. I was surprised that the admission of Maine was being talked about so much as early as the 1780s; maybe that just means I'm ignorant.) In February 1782, Vermont's legislature complied, but objections from New York still prevented Vermont's admission.

Vermont took the position that it was not a part of the United States and was at liberty to enter into separate peace negotiations with the British. The negotiations took place in Quebec and Vermont, and resulted in numerous prisoner exchanges and in 1780 resulted in a secret armistice that held until the end of the war. Vermont's diplomats had secretly promised the British that a political process was being set in motion to convince Vermont's legislature to make Vermont a British colony. In October 1781 a Vermont soldier, Sgt. Archelaus Tupper, was killed in a skirmish and Colonel Barry St. Leger of the British army ordered him buried with full military honors and wrote a letter of apology to Vermont's General Roger Enos. The letter was leaked and a scandal broke out over Vermont's "treasonous" negotiations with the British.

In the peace treaty of 1783, the British ceded Vermont to the United States, but Vermont continued to maintain that Vermont was not a part of the United States.

In 1784 New York's governor, George Clinton, wrote that he would use military force to overthrow Vermont's government if Congress didn't authorize using federal troops for that purpose. But he did not do so and Vermont continued to exist as a de-facto independent country.

In 1790, New York's legislature consented to "the community now actually exercising independent jurisdiction as 'the State of Vermont'" becoming a separate state within the Union, provided the boundary could be agreed on. Vermont's negotiators insisted on also settling the land disputes rather than leaving those to be decided later by a federal court. The settlement was for 30,000 Spanish dollars, where New York had asked for $600,000. (The U.S. dollar did not yet exist. Where the Constitution of the United States, in two places, refers to "dollars", that meant Spanish dollars.) Alexander Hamilton of New York had led the movement to let Vermont go. He wanted more northern representation in the U.S. Senate.

The first act of Congress admitting a new state to the Union was passed in February 1791 and admitted Kentucky, as of a specified date well over a year in the future. The second one was passed two weeks later, also in February, and admitted Vermont as of exactly two weeks later. That bill said that "the State of Vermont" had petitioned Congress for admission, in effect acknowledging that it was an already existing polity. Unlike what happened in many states admitted later, no new state constitution went into effect at the time of admission, and the officers of the state (governor, governor's council, judges, sheriffs, etc.) simply continued their terms of office that were already underway under the 1786 constitution of Vermont (which had superseded the 1777 constitution).

So Vermont was considered for well over a decade, by the state of New York, to be a district in rebellion against New York.

I have before me Robert Mello's book Moses Robinson and the Founding of Vermont. That will tell you all of this. See also Ira Allen's Natural and Political History of Vermont and Frederic Van de Water's The Reluctant Republic. There's also a lot on this in Benjamin Hall's very thick book History of Eastern Vermont.