I really dislike water rides, but I "take one for the team" and go on them because my D and H love them. So here is my strategy for staying dry:
1) If it's hot that day, I will wear something on bottom like those quick-dry Nike running shorts, and a short sleeved tee or tank top (as a base layer if it's chilly). If it's cold, I will wear something like leggings or sweats on bottom that I can push up to knee-level or above.
2) I bring either a poncho or a tall kitchen garbage bag with armholes cut out (it works because I'm only five feet tall), along with several large ziplock bags and a pair of flip flops, in a backpack that day (if I'm lucky my H is bringing one anyway)
3) Bring a small towel/paper towels/napkin in the backpack too
4) I don't bother to spend a lot of time on my hair that day and will wear it in a ponytail/braid and/or with a baseball cap
5) I bring an extra baseball cap or borrow my husband's at the crucial time (see #9)
6) Before getting in line, I take off my sneakers and socks, put them in a Ziplock, put on the flip flops, and put my purse in the backpack
7) Right before I get to the final queueing area, I take off any layers over the tee/tank and put on the poncho
8) Before I sit in the ride, I wipe down the seat with the towel/napkin and be sure to tuck my poncho/trash bag under me; push up leggings/sweats as necessary
9) I take the spare hat and cover my face with it when we get near any parts of the ride that might splash my face (like the big downhill or near that fountain that people have mentioned)
Ok, this routine may sound ridiculously high-maintenance to some. And I never did any of these things until I rode once and got absolutely drenched...seriously, it was like the scene in Fantasmic where Mickey keeps getting doused with bucket after bucket of water. It was insane...every last inch of me was drenched, makeup was running in my eyes, and I was wearing jeans, which feel extremely heavy and uncomfortable IMO when they're wet (and caused really bad chafing). This routine may also seem very vain to some...but I have severe rosacea that takes a long time to cover every morning and is very embarrassing to me, so I would rather hold that baseball cap over my face to prevent all my makeup coming off then have to feel self-conscious all day or carry around my concealer with me and re-do it in a park bathroom.
And my hair gets super frizzy and out of control when it gets wet...hence the ponytail/braid/baseball cap.
And I do it all with a smile (that's admittedly sometimes hidden by the baseball cap!) and feel lucky to be enjoying WDW with my H and D.

Oh, and this strategy works pretty well for Kali River Rapids too!
Hope my strategy is helpful to you, OP!
