Abby, perhaps an opinion from a total outsider (I´m a "large choir" tenor just learning traditionals) could help.
You say that you are not a good singer. Most probably you are wrong, but let´s assume it´s true. Then you say that you sing a capella these songs. You get too exposed. Even a minimal flaw will be noticeable, and the audience will surely not get into the mood. I would rely on an accompaniment, it will help you maintain the pitch and expression, and if he/she is a good musician, will make you feel comfortable and secure.
I would also quote Michael K.:
Perhaps, tape the tune, and have a few people who's opinions you respect and trust, and who can be (brutally if need be) objective, to listen to it, and get their feedback.
Don´t give up !

Capt.Bob, my two dogs listen to my songs as long as they receive a cookie, I´m now reducing rations and some day I´ll get them listening for nothing. Alas, I´ll never make them pay me.
Un abrazo - Andrés (from Buenos Aires)