What would really help is to write some poetry
What trying to write structured poetry does is force you to put a lot of thought into how you build sentences in order to fit a predetermined rhythm and/or rhyme formula. Quite often, you have to get very creative about how and where you start and end sentences, and everything in between. Doing so is excellent practice for structuring prose.
Take, for instance, The Raven by Edgar Allen Poe. It's moderately long as far as my attention span goes, but even if you don't read all of it, just try reading a verse or two out loud. Listen to how the lines flow into one another. Listen to how the pattern of stresses make the poem bowl along at pace, then pause and ramble, then bowl along again. Poetry teaches you how to craft a sentence that sings. How to use rhythm and stresses to evoke different sensations in a reader.
This is valuable beyond simply learning novel ways to start a sentence. It teaches you novel ways to write the whole thing.
How to start
Of course, it's tricky to write poetry if you don't really like reading it. It may well be that you don't like poetry full stop, but it might just be that you haven't found a poem you like. I'm relatively picky (some might say uncultured) with the poetry I like, so it takes a while to find things.
Worst case scenario you can treat it as a practice exercise. Here's what I did.
Pick one verse of The Raven (or any poem you like). If The Raven is a bit complex, I also like The Conqueror Worm (the second-from last verse is especially good). Read it and work out the pattern of syllables to each line. Then work out the pattern of rhymes if it has one. Then work out the pattern of stresses.
Next is the tricky part. Rewrite that verse on a topic of your choosing. Force your sentences to fit the structure of the poem. Formulate and re-formulate until you have a verse that flows as well as the original, just using different words.
It's a bit of a trial by fire, but at the end of it I had real appreciation for good poets, and a bit more experience at novel sentence structures.
Do it enough times, and it starts to become second nature :)
tl;dr You will probably gain more from trying to write poetry than just reading it, and can treat rewriting an existing poem as a practice exercise for writing prose.