The following are mostly of European-American and/or much-diluted African-American origin and are, at least in my experience, usually moderate-to-fast tempo. Not all of them are particularly old, though (but then we're not that old a country, are we?). It's a bit of an odd question, though, because the distinctions between "singing" songs, dance tunes, fife tunes, etc. get blurry very fast, as does the "ancestry" of many songs, since many "white" folk tunes started out as published sheet music or were restructured from African-American tunes (Old Joe Clark, Little Liza Jane, etc.).

And I'll warn you that my experience is mainly in string-band music, which is less likely to be sung and more likely to be played very fast, so I don't think folk music is particularly slow. From what I know of the ballads that are most near older British tradition (Barbara Allan, Barkshire Tragedy/Two Sisters, Lord Bateman, Old Bangum, etc.), yes, they do appear to be in the slower-to-moderate range, but it's hard to sing intelligibly much faster than that, and I don't know how British our versions of these still are.

Dance tunes that are also commonly sung:
Sugar in the Gourd
Cotton-Eyed Joe
Barlow Knife
Way Down on the Old Plank Road
Train on the Island
Ida Red
Sally Anne
Wildwood Flower (ca. 1859--does that still count?)
Dixie (ca. 1858. Sometimes played as a dirge, yes, but sounds better fast)
Kingdom Coming (Year of Jubilo, probably ca. 1866)
Fall on my Knees
Danced All Night With a Bottle in my Hand
Paddy, Won't You Drink Some Cider?
Cluck, Old Hen
Waterbound/Stay All Night

Songs (not generally danced-to):
Wild Bill Jones
Been All Around This World (Hang Me, Oh, Hang Me)
Bring the Traitors In
Girl Behind the Bar (West-Side Tavern)
Greenback Dollar (similar to Gold Watch & Chain)
Johnson Boys
Little Sadie
Old Bangum (the one about the killer hog)