Dave, I do something similar when singing on long car trips, which I've always wanted to try in a group. This may also answer Sophocleese's concern. 1) I sing a song, whatever I think of and feel like singing. 2) I sing a song that shares a WORD - just a word (no fair using "the" or anything like that unless you're desperate or suddenly really want to sing a totally unrelated song) with the prior song, and so on, chaining by free-association. That way I don't spend any real time, maybe a minute or so, thinking of the next song. So I could sing The Wreck of the Old 97, then another song that mentions, say, throttle (in any meaning) or husband (can't think of any I want to sing right now) or ladies (ok, I'll sing Come All Ye Fair and Tender Ladies... then Strawberry Fair.... and so on. The word doesn't have to be in the title or even a main part of the plot).

Footnote, prompted by a shared desire to be honest and name-drop: I mutated this idea from one of AllanC, from back before I knew him through the Mudcat, who told me once he used to challenge the audience to say a noun and he'd sing a song with that word in it. The only word I can't think of a song about is davenport - I got sofa and easy chair and couch, and even ottoman, but no davenport. I'll spring that on him when he gets back.
If the trip is really long, I might sing all the versions I know of one song before going onto the next, which can then take only a word that appeared in ALL versions of, say, Soldier Soldier or the 2 sisters or the 2 brothers or whatever. I think all of these would be fun around a campfire...