Some years ago I did a frontiersman program in the Boy Scouts of America, and learned history and skills from the American Revolution. Some nights we heard songs from the era that revolutionaries sang. This one in particular was a father answering questions to his son as Washington's army marched by.
I remember a common verse in it was "that my dear son" and another was "that old british rag" (in reference to the British flag when the son asks about the Patriot's flag)
I have looked online in databases of songs from the period and cannot find anything.
I also apologize if this post is out of place for this subreddit, but I really couldn't think of another place to find this information.
It could have originated on some obscure broadside that I don't recognize (admittedly, I know far more about British broadsides than American ones) and that hasn't weaseled its way online in some form, but I would hazard a guess that it was a recently composed song by one of the performers. (Nothing you recall about the song strikes me as particularly indicative of an older text—if anything, quite the opposite.) Do you remember anything about the identities/affiliations/musical styles of the singers?
If you don't find an answer here, you might also try /r/tipofmytongue