How do I change date for my pet without cancelling the entire stay?

There should be and EDIT option beside each date. If an owner doesn't have the option, the sitter has. I have used it myself to change dates when an owner's plans changed between the original request and the booking. I have also used it to extend a stay for an owner who was unable to travel home due to bad weather.

You'll have to contact Rover support and have them fix it from their end.