Is it wrong that I think the frenemy-ship I have been in with my man's ex is like Dolly's song Jolene, and I'm the girl?
So you think that you're Jolene?  Do you know what happens to her in the song?
It's just like I'm always trying to prove myself better than her
It can be nice to notice parallels between your life and art, but nobody expects you to always be friends with your partner's ex-lovers.  And besides, even if you think the song is similar to real life, you shouldn't deliberately try to make things happen the way they do in the song.
Why not?
Your experience is different from that of the artist. You need to conduct your life in your own way.