How do I do this without getting the book soiled? The only solution I can think of is to wear plain white cotton gloves.
Get a Kindle.
And what’s the problem with that solution?
If the problem is that cotton gloves don’t grab paper very well, they make beaded gloves which have rubber dots on the fingers for better grip.
Other options I can think of are: e-book readers, reading cheap paperbacks that you don’t mind getting soiled, reading glossy magazines (which hopefully won’t absorb moisturizer), and nitrile gloves.
I thought this was going to be a braille thread. 
I was uncertain if there would be enough friction to turn pages. But your suggestion of beaded gloves solves that. Thank you!
FYI, disposable Nitrile exam gloves also work. Basically very thin rubber gloves. Your hands may get damp and sweaty, but that’s not a bad thing if you’re trying to keep them moisturized. Latex should work too. Don’t get vinyl gloves, those are slippery.
They sell cotton gloves intended to be worn at night with your hands heavily moisturized – I picked some up in a drugstore once, although I don’t use them often because the “one size fits all” didn’t fit me that well. But I have read books with them on. It’s a little harder to turn the pages, but not a major issue.
FWIW I recently started using the CVS store brand version of Vaseline Intensive Rescue lotion, and it seems to work well while also not leaving your hands greasy for very long.