So I'm trying to Adult and was wondering how to set up utilities at a new place while keeping them running at my old one?

I have a decent amount of overlap (1 month) between the places, and I think I legally have to keep the utilities on at my old apartment, since it's still leased to me though it will be vacant, while I move into a new place and start up new utilities under my name. It would be the same companies, as I'm only moving a few blocks away. Can I just add a new address online and start those utilities and then stop my old ones in 3 more weeks (I'm already 1 week into the overlap). Ugh, this is so confusing. Any insight would be much appreciated!