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

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

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.