I am sensitive to spinning, it makes me feel sick. Even things like the Toy Story Midway Mania (or whatever it's called) ride at Hollywood Studios can make me sick because the car will spin fast to face the other side sometimes.
Which rides at the parks have a lot of spinning?
I'm excluding the roller coasters since those don't spin per-se but you'll have to decide if you like those or not.
In IOA:
Storm Force Accelatron is all spinning. It's the teacup ride.
Spiderman the vehicle spins 3 or 4 times.
Forbidden Journey doesn't spin all the way around but it twists you back and forth.
Cat in the Hat spins you around multiple times. This is similar to the Toy Story Mania ride you mentioned without the shooting.
Caro-Seuss-el is a carousel and One Fish, Two Fish is like Dumbo. They both turn in a circle but your car isn't spinning.
Studios:
Mummy spins your car once or twice.
Gringott's spins the vehicle a few times.
Men In Black spins you around multiple times. It's the same ride car as Cat in the Hat and also similar to Toy Story Mania. It is a shooter ride like Toy Story.
Kang & Kodos is like One Fish, Two Fish and Dumbo.
Transformers your vehicle spins 3 or 4 times. It's the same ride vehicle as Spiderman.