You should be able to charge things to the room with cash you've given them ahead of time. There are a few places where you can't use your cart, and a handful of them are actually snack stands that don't move (like the Beaver Tail stand), but you can use it almost everywhere.
The limit isn't $3000 anymore. Instead, they run a batch transaction on your credit card when you reach a certain point, depending on what hotel you're staying in. Then you start back over after that, so there's no upper limit.