I think 'Guest Russ' meant that the Carter family are definitely not traditional British folk, not that 'Guest Emmie' should definitely not be looking for traditional British folk. The possibilities for confusion here are endless ;-}

Assuming Emmie is in the UK, there are loads of gatherings during the summer where people get together, sing and swap songs. Most festivals have a 'fringe' where you can experience sharing songs within a group (rather than having songs sung to you, which is a totally different thing). Visit a few and pick up any songs that appeal to you - you can come back here for background info, words etc. of anything you like.

Sadly, life is too short, and the internet too limited, to list any more than a tiny fraction of the songs that fit your criteria, even if we do limit ourselves to traditional British folk.

Best wishes anyway.

Andrew