John,
Apart from trawling through endless manuscript collections and indeed published accounts of the period, your best chance is looking at what survived to be collected and the differences between American versions and British versions, and you would do really well to co-operate with Richie in this venture as he has already made a strong start.

You are right to suggest the most likely sources of material migrating is oral tradition and street literature. A strong source seems to have been those little songsters with about 200 songs in published on both sides of the pond in the early part of the 19th century. Sharp certainly believed when he was collectiong in the Appalachians that many of the ballads had been circulating there for more than a century. There are indeed a few early ballads that were found in America that had long died out in Britain.

FWIW you need to be aware that literary interference existed both this side and your side. From an early stage collectors were fabricating and expanding oral material, for various reasons, mostly commercial. And indeed there is evidence to suggest that some of these fabrications went into oral tradition. Equally a similar process was going among the broadside hacks who saw nothing wrong in rewriting an old ballad to turn a quick shilling. This process often accounts for those ballads that have widely varying versions.