Hi Cassie, I've been writing folk songs for many years and I've had some success with them as well. Some of my songs have been recorded by other artists and 9 of my songs have been signed to music publishers. Some of my songs are currently being submitted for possible placement in a movie.
You certainly don't have to rhyme your lines all the way through the songs. If a song is well written and communicates something to the listener there are many people that will be interested. As it turns out, the songs that I've had success with have all had rhyming lines. Most folk singers (and folk music publishers) want a well structured song (verses, chorus, bridge) that rhymes.
I don't want to discourage you from writing songs that don't rhyme, I just wanted to pass on my own experience. You might just find a publisher or artist that loves your creations as much as you do. That's what makes this business so great.