will a pet sitter stay overnight in my home?

Yes, that is a service many sitters provide. it is called Housesitting. You can do a search of sitters in your area (by zipcode) and filter according to your needs. Just be sure to search for that type of service offering. Then contact several and meet with them to discuss your needs. I can't give you any types of specifics as to the number you will find because I have no idea where you are, but it is a fairly common service.

If you don't find any, there is always a drop-in service, in which the sitter visits your home several times a day to take care of your dog (walk, feed, play, etc.); it is on a per-visit basis. If all else fails or you are having difficulty locating that type of sitter, you can always call the Rover Customer Support Team at [Edit: Rover’s contact options have changed. Visit the Rover Help Center at https://support.rover.com/ to find the phone number, help articles, or chat with the team] for more personalized level of service.