I find that there are a few singarounds who are quite happy for you to play a tune instead of singing when it's your turn. In an evening, if I get two turns, I will usually play a tune in one and sing a song in the other and that seems to go quite well at Folk Clubs I go to

Yes, and if you want to play with others you can usually invite others to join you when it's your turn. That's fine, as long as they don't think they can join in with everyone else, every time (there was another thread about session/singaround etiquette recently which discussed this sort of thing).