Yes, there can be stroller theft, so don't leave things like wallets, purses, the Hope diamond, etc in them.
But do put identification: bandana or something bright on them so you can see it. The biggest issue I have is trying to find where those sneaky cms put it! There is designated parking by all the rides or lands, and cms will rearrange the strollers into neat rows. If you still feel insecure, get a stroller lock.
At TS places there is also parking outside the restaurants. But I've never had a cm force me to park the stroller and remove a sleeping child from the stroller. So if munchkin is snoozing hardcore, they let you in with stroller.
With regards to diap bags, may I suggest SkipHop brand? Their bags look like purses, so you don't feel weird carrying a giant obvious bag. Or you could look into getting a simple diaper wallet. Both save space in different ways.