You are confusing two sides of music and looking for a bridge between them as if Classicsl led to jazz and you're trying to find the missing link. Each had its own evolutionary line. American classical music was trying establish itself, but the classical music in America in the 19th century was mostly coming from Europe, although there were some American virtuoso players who were making an impact and that eventually led to some American composers.

On the other hand, throughout history there has always been a thread of popular music separate from classical. Street musicians, troubadors, tavern players, minstrel shows, etc., are all examples of popular music styles. Much of this music from the pre-recording era has been lost because it was passed on orally and seldom notated on paper. In the mid to late 19th century Stephen Foster was one of the leading songwriters. Camptown Ladies is probably his best known song, but the 90s Appalachian Spring featured two of his most beautiful songs - Hard Times (sung by James Taylor) and Slumber My Darling (by Alison Kraus). This songwriting tradtion continued on into the 20th century and blossomed during the vaudeville era with composers such as Irving Berlin and Zez Confrey, and as Vaudeville morphed into Broadway, George Gershwin.

Jazz was a continuation of the African music brought by slaves from their homelands. In the fields they would strike up a rhythmic song to pace the workflow and pass the time. After the Civil War these rythms became more organized and around the 1890s became ragtime and was written down. There is a bit of lost history here because the black music of the post-war era wasn't recorded either on disc/cylinder nor notated on paper until ragtime came along. By the way, many of Stephen Foster's songs were written for minstrel shows, and some tried to argue in the past that the music represents some kind of notion of what black music and dialect must have sounded like, but that is widely discredited today. It would be like trying understand human anatomy by studying a Simpsons cartoon. It probably wouldn't even be that close. Minstrel show music was what white racists thought black people sounded and acted like, exaggerated for what passes for comedic effect for racists.

At the same time as ragtime was becoming popular (best exemplified by the incredible music of Scott Joplin) jazz was developing in New Orleans by people such as Buddy Bolden and a youngster named Louis Armstrong. The earliest jazz was a free-for-all style known as Dixieland, but by the late 20s was starting to organize into a large band format that by the 30s came to be known as swing, an evolutionary step also led in large part by Louis Armstrong. The success of jazz was largely because it was getting recorded and distributed around the world, and jazz was the first music that America could lay claim to inventing.

George Gershwin was determined to create the first real American classical music style by combining jazz (recognized around the world as a distincly American musical flavor) and classical, and for the most part he was extremely successful, with works such as Rhapsody In Blue, An American In Paris, and Porgy & Bess being widely praised during his lifetime. Unfortunately, his lifetime was extremely short, and he died in his 30s and the world was deprived once again of a truly musical genius.

TL;DR: There was always music there before jazz, it just wasn't widely recorded in any way, so it is sort of invisible to history. Classical music had a separate bloodline. The two didn't mix until after both were well-established as separate genres in America.

In New England, a genre known as "contra dance" (a variant of "country dance", not of "against") has a long and distinguished tradition going back at least to 18th-century England and France, and possibly before that. It was essentially music for dancing, and in particular music for dancing with a partner.

Music travels across political and cultural borders. People from England, Scotland, and Ulster brought their music with them as they settled throughout the eastern United States. People of African descent had their own traditions. And they listened to each other. Even people who probably wouldn't talk to each other on the street even if they wanted to listened to each other's music. With only some exaggeration, if you want to hear a quick tour of what American popular music might have sounded like in 1920 or so, listen to Jimmie Rodgers, the Singing Brakeman. Hearing Rodgers isn't like listening to some quaint, authentic, boring art form that ain't no fun but is good for you; he was a great songwriter and a great singer. He was aware of Appalachian music, of blues, and of folk music, and took his synthesis in a very different direction from what Louis Armstrong was doing -- with some of the same elements -- at about the same time.

Dig around, and you'll certainly find other local and regional analogs to our contra dances. "String bands" and what's now called "old-time music" are good places to start. (I enjoy the Carolina Chocolate Drops.)

One of my favorite books about musical synthesis is Kentucky Country by Charles K. Wolfe. It covers a more recent period than what you're looking for, but it's fascinating about cross-cultural pollination, made maybe even more so by Kentucky's somewhat anomalous status as not quite Southern, not quite Midwestern, and certainly not Eastern.