I'm not a "best rhymster," so I'll just say what it's like for me.
It's actually a very strange and ambiguous process. When I get to the end of a line and know that I'll be called upon to rhyme the final word two lines later (for example), I'll "commit" to the final word without knowing exactly what the next rhyme will be...but it's not as if I'm ignoring rhyme, either. It's more like I try to get a "feel" for whether the word (a) will have enough stress to be heard in retrospect, when the rhyme finally happens, and (b) will yield rhymes that are in the ballpark of where I feel the content of the poem is moving.
Sometimes, when I get to the line that's supposed to rhyme, I encounter difficulties and I might go back to the first line and recast it for a different rhyme. And sometimes I can anticipate a rhyme that I may want to use and I try to adjust the content to lead up to that rhyme in a way that seems natural (i.e., not rhyme-driven). When that happens, sometimes I find that the rhyme word I'm interested in using forces me to come up with images or metaphors (or other content) that will allow that word to be used, so in a sense the content does become rhyme-driven...but as long as it doesn't "show," I don't see that as a problem but as one of the pleasures of writing in rhymed meter.
I suspect that lots of aspects of composition, and not just rhyming, work in this way. The poet writes a line based on "feel" and then tries to keep the action moving in a way that's faithful to the line that's been written, but often the poet needs to go back to revise the line that's been written to conform to the new lines that were written later.
I'll stop there, since I'm growing increasingly incoherent.
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