I don't have DID, but I definitely get this after both therapy and sessions with my case manager, or even just after really stressful events. It's really energy draining physically and mentally to experience such strong emotions. That's actually normal.
For me, I make sure not to plan anything after on the day of. If possible, I try not to plan anything the next morning as well, so I can take the time to rest and not have to worry about anything besides resting. I know not everyone has that luxury, but if you can shift things around to minimize what you do on those days, it might be helpful.
Also sorry if obvious, but definitely ask your therapist for suggestions as well. Maybe they can help spend a few minutes at the end of sessions working with you to try to calm down some, or have some suggestions for other ways to take care of yourself and rest if you still have other things you need to do. Or even work with you on finding safe ways to release your emotions outside of sessions, so you don't have to let things build up so much.