Have American writing styles changed between the Civil War and Today?

I'd guess that it is mainly selection bias. We have all heard of Mozart, Liszt, Bach, but never heard of Salieri, Emile Bernard, or Khosrovidukht. Similarly, we have heard of Pachelbel's Canon in D, but not Als der Gütige Gott even though they were both written by Johann Pachelbel. Art tends to survive only when it is good.

It is almost certainly selection bias. In fact, a dialect humor that resembles today's "Lolcat" was all the rage as comedy and Lincoln used to read it before cabinet meetings:

HIGH-HANDED OUTRAGE AT UTICA - Artemus Ward

In the Faul of 1856, I showed my show in Uticky, a trooly grate sitty in the State of New York.

The people gave me a cordyal recepshun. The press was loud in her prases.

1 day as I was givin a descripshun of my Beests and Snaiks in my usual flowry stile what was my skorn disgust to see a big burly feller walk up to the cage containin my wax figgers of the Lord's Last Supper, and cease Judas Iscarrot by the feet and drag him out on the ground. He then commenced fur to pound him as hard as he cood.

"What under the son are you abowt?" cried I.

Sez he, "What did you bring this pussylanermus cuss here fur?" and he hit the wax figger another tremenjis blow on the hed.

Sez I, "You egrejus ass, that air's a wax figger--a representashun of the false 'Postle."

Sez he, "That's all very well fur you to say, but I tell you, old man, that Judas Iscarrot can't show hisself in Utiky with impunerty by a darn site!" with which observashun he kaved in Judassis hed. The young man belonged to 1 of the first famerlies in Utiky. I sood him, and the Joory brawt in a verdick of Arson in the 3d degree.

It is not selection bias.

Types of writing should be compared only against a comparable type of writing, controlling for the education and social class of the author, using comparable methods to a proper cross-sectional analysis. This is commonly done and then the text is placed into a software program to create a text analysis, so there this an "objective" answer to this question, but we simply do not know what it is at this time. For example, a speech by Abraham Lincoln could be compared to a speech by Barack Obama; a letter written by Sherman to a letter written by General Petraeus; a letter written by a low ranking army conscript with little education to an email written by a modern high school senior. I feel certain that the differences in style would remain quite striking. This should not be surprising, since language, art and writing styles change over time due to social influences.

No one today would say, as per the OP:

Do not hurry, men, and fire too fast—let them come up close before you fire, and then aim low, and steadily.—Gettysburg, p. 436

(For non-native American English speakers: several parts of this are off-putting. "Do not" as a command, "hurry" as a verb, "men," and "steadily" sounds very exact. Taken together, this phrase is highly unlikely to be uttered by a modern American)

Now, as Oldcat points out, Civil War humor is largely left out of modern historical accounts, which represents a selection bias. This has occurred because either the humor would not be understood by a modern audience or it is a dark humor, which is arguably no longer socially acceptable in most cases. Quoting major historical figures using dark humor has the effect of making them seem strange, twisted or callous, rather than heroic and ennobled, which is the effect in much of Civil War era writing. Nevertheless, it still often relies on classical literary device and the changes in uses of humor do show a change in writing styles overall.

There are mainly three reasons for the simplification of writing styles (but there are more as well): 1) changes in art movements 2) changes in education 3) changes in views about the audience. The third point runs into the first two points, but I see it as the most important so it will be examined.

Art Movements

The time period around the Civil War is heavily influenced by Romanticism.

The movement validated intense emotion as an authentic source of aesthetic experience, placing new emphasis on such emotions as apprehension, horror and terror, and awe

Writing styles from this time period thus appear much more "poetic" in nature, but it was soon replaced by Realism. Much of modern writing is probably properly called Utilitarian. It is also Postmodern and many American writers still rely heavily on Realism, even in more creative writing forms.

Education

Classical education ended in the 1960s. One of the main reasons is that the sciences and "social sciences" became much more important. You are correct that Americans in the Civil War who were educated, which is a much smaller percent than today, would have had a much stronger background in literature.

By the end of the 18th century, in addition to the trivium and quadrivium of the Middle Ages, the definition of a classical education embraced study of literature, poetry, drama, philosophy, history, art, and languages.

They would have learned Latin and Greek and learned the classical structures for writing speeches, such as Eulogy or Apologia. The Gettysburg Address is an example of a Eulogy.

Attitudes Towards the Audience

The most important difference is the way the creators of books, speeches and historical accounts view the people that will read it. According to Dominic Strinati in "An Introduction to Theories of Popular Culture" there are three ways to divide art by its audience: folk art, high art and popular culture. In the late 19th century, art movements started to address art to be created for the "peasants" or the "volk." The early 20th century included many disputes about mass media creating low brow mass culture.

Today, more or less popular culture has been victorious. A major reason is that mass media markets expanded in the 20th century and we are continuously exposed and normalized to pop culture messages, but it is also simply a cultural shift. Writers are encouraged to keep communications simple and understandable to everyone in society, unless a book or communication is absolutely directed for a small number of experts. Political speeches are no longer flowery or use complex grammar, because it is considered more important that everyone understands and participates in the democratic process. Letters are largely not written at all and have been replaced by email, text messaging, instant messaging, or even picture messaging.

The "goodness" of a writer is now measured by how well they communicate a message (so much so de Toqueville is being called a bad writer!?). During the Civil War, writers would have been much more concerned with the style of their writing. Style can also be used to communicate messages.

I don't think your question is answerable in any objective sense. There's no way to unambiguously measure the "goodness" of a writer. That being said, subjective and anecdotal experience does have some value.

Selection bias and association probably have a lot to do with what you're seeing.

Consider how people often complain that music today is shallow and pointless, and that music of an earlier generation is so deep and well-crafted. Every era has music that's cheap and frivolous (Roger Miller's Do-Wacka-Do)*, it's the good stuff that survives the longest.

Let me quote a passage to you from a well-known writer of the 1800s:

As to the influence which the intellect of one man may have on that of another,
it must necessarily be very limited in a country where the citizens, placed on
an equal footing, are all closely seen by each other; and where, as no signs of
incontestable greatness or superiority are perceived in any one of them, they
are constantly brought back to their own reason as the most obvious and
proximate source of truth.**

I understand what the author is saying -- and he says it quite precisely -- but his writing style seems poorly organized and overly profuse.

Here's a sample from another author of the era:

The road is between this and the river, all near together. We go into the water,
then stay out till our clothes get dry and jump in again. Lots of the boys are
at work with a seine catching fish. They are very poor biters and never try to
get away when caught - no game at all. Very bony and soft. The natives call them
hardtails. Another kind humpbacks and flatheads. We stayed in the water nearly
all the time.***

The contrast between the two authors is strong. This one seems less-educated, a man of simple letters, narrating a simple story of his own experience. The word choice and phrasing seems almost quaint in a way.

In my opinion, neither of these two writers would be considered to have good writing style today. But with the passage of time, writers like these have been passed over (at least for their style), while writers like Mark Twain have been read and re-read by each generation.

*Though I do happen to like Do-Wacka-Do. http://www.youtube.com/watch?v=UI-Y0CMGwxo

**Alexis de Tocqueville, Democracy in America, 1835

***George Hand, Diary of Military Service in the Southwest, 1864