You probably don’t need to be locking up the rears in a rental kart. If you’re driving that hard into the corner, that could be why you are seeing the kart hop as well.
You should almost always be leaning back in the seat, in any kart, so you can get the full effect of the weight transfer.
Shouldn’t be sliding at all, so if whatever you’re doing with your posture is causing the kart to slide, you should be trying something else.
Here’s another good topic on posture that covers a lot of this: Posture, and head movement