Yup. My understanding is that the new building will be complete Spring 2023.
Drexel runs a free bus service between Queen Lane, Center City, and the Main Campus, so you could theoretically choose to live in any of those places and commute for free to either campus if you don't want to move. For various reasons, Center City is a great choice if you don't mind a free bus commute. I have no idea whether bus service to Queen Lane will continue to be offered once the new building opens, since I think the plan is to eventually sell Queen Lane.
If it were me, yeah, I'd make my life easy and plan on moving after M1, unless you want to be in Center City.