There are loads of sessions where people, experienced or not, only get one song. I would prefer to hear one wonderfully done song from everybody than two or three less well performed.
I will always prepare one song but have a couple in reserve in case it is quiet night and there are fewer singers who need to sing two or three during the evening. I know a number of people who go out several times a week to different clubs in order to be able to sing more songs but this can lead to divorce, as you so rightly pointed out.
Just keep singing and over time you will build up a good repetoire of songs you can, not just sing, but perform well. When that happens you will be a name too.