Hi, Jennifer!
Take a look at this great post about
Rider Switch at Walt Disney World, and you'll see that Rider Switch is offered at select Walt Disney World Resort attractions. In those instances (such as The Barnstormer), the wording has been chosen very carefully: "If Guests in your party don't want to board an attraction, you don't have to miss out. One adult can wait with the non-rider while the rest of the party experiences the attraction without having to wait in the regular line again when another adult in the party returns to supervise the child..." You'll note, though, that most of these Rider Switch attractions are the more intense, height-restricted variety, rather than the spinning rides.
My husband doesn't do well with spinning rides either, so when the kids were smaller, I'd be the one to run each child onto the ride if we needed to ride in pairs. In those instances, we'd all wait in line together; when it was our turn to go onto the ride, I would simply explain to the Cast Member at the front that my husband can't ride, but that each child needed supervision. I'd ask if he and child #2 could just wait there at the front (as if we'd asked for front row in a roller coaster, stepping aside to let others behind us onto the car so that we could wait for the next) until child #1 and I returned. This was never a problem.
Keep in mind, though, that some attractions' setup make this easy (such as Prince Charming Regal Carrousel, where you can see the entrance and the exit from the Carrousel), whereas others include an exit which is not at the entrance. In those cases, simply ask a CM as you enter how they might go about accommodating your particular needs. I'm sure they'll happily offer you a solution at each and every ride!
Have a wonderful trip, Jennifer! Send more questions as you think of them; we love to help!