I know exactly what you're talking about. I would say that it's very rare for the lyric to exactly compliment the original 'idea' of the music, at least when I'm writing songs.
But I wouldn't say there's a battle between letting the music or lyrics win.
Once the music is written, it needs a lyric to be complete. If you compromise and settle for the wrong lyric, your song is in jeopardy (IMO). But a piece of music is more resilient than you may think. Once you have a lyric that works, and feels honest, you'll be amazed how flexible your piece of music will be as you make subtle changes to accommodate the words.
Sure, it will be different, but it will be the same piece of music.
If you're looking for advice, I would say this: don't separate the music and the lyrics in your mind. Once you have a melody and a rough arrangement, don't think of it as that part being 'finished'. It's a framework to hang your lyrics on. And how they turn out will effect the way the music part of the song works.
Step back and allow the music to be 'in progress' until the words are finished, at which point the whole song can be finished as a single element.
How you decide where the song goes is personal to you. I generally find that I need a strong idea as to what the song is 'about' before I can finish it. Otherwise I'm just clutching a slightly abstract lyrical ideas and trying to fit them into my musical idea. This works once in a while, but they tend not to be my strongest songs.
I hope some of that helps