1. You don't have to sing. Many, many great improvisers don't like the singing part.
2. An improvised song cannot be as good as one written by a professional songwriter. So don't expect it to be.
3. Keep it simple. The songs that you end up singing all day are not usually the most complicated ones.
4. It is acceptable to twist a word to make it rhyme.
5. Most popular songs include lines that do not rhyme.
6. All technical faults can be forgiven if you sell, sell, sell the song.
7. Improv audiences actually prefer someone with a bad voice singing with gusto than someone with a great voice singing effortlessly.
8. Feel and sing the emotion.
9. The audience would generally much rather hear someone belt their simple heart out about their love for the other character than hear a long list of cleverly rhyming innuendo. (Which doesn't mean they won't enjoy any of the latter slipped in the former.)
10. A good pianist is the secret to a great improvised song.
11. Allow yourself time. Let the first couple of bars go by so you get the tempo. Believe me, singers rarely start singing on the first note. If you feel that this is dead time, then posture and move about like a singer preparing to sing.
12. There's a reason pretty much every song has a chorus.
13. Someone else will almost certainly want to sing if you don't.
14. Practice in the shower. If you are still too shy to sing on stage, invite increasing numbers of people into the shower until you feel confident enough to do it on stage.
15. Repeat. For God's sake, repeat.