I went at 39 weeks to a week-long camping event an hour from home. I took a proper mattress to sleep on in my tent, and didn't have to do a whole lot of walking because the heads of the event gave me priority rides on the golf cart shuttles the whole time (I was on the volunteer staff).
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FWIW, I was diagnosed with polyhydramnios late in my first pregnancy; it was the excuse my dr. gave for inducing at 41 w 6 days, and in this one when I went in for a sonogram at 40 weeks to confirm position. It wasn't severe, and has obviously not affected the baby's growth (DD was 7 lb. 13 oz., and last week they estimated this babe was almost 9 lbs.). It obviously didn't result in preterm labor for me, either. The only risk of polyhydramnios I wound up with was a cesarean, and that was the Dr.'s fault. My midwife doesn't seem to think it's a concern.
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Going into labor at 37 weeks generally isn't considered preterm; I'd just be prepared to head home if you think you're going into labor. That is, if you haven't had very short or early labors with previous pregnancies.
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If it was me, I'd go ahead. In the end, you need to trust your own intuition. If you need reassurance, get a sonogram before you go, just to make sure the baby's still doing fine.