Hi ibroker,
Thanks for coming to Apple Support Communities for help today with Handoff on your MacBook Pro. It sounds like it won't work between your iPhone and the Mac until you quit iMessage.
Have you tried any isolation steps at all? I'd suggest checking this issue out when booted into safe mode: Use safe mode to isolate issues with your Mac. This can let you know if a third-party app might be causing the issue.
Also, test the issue inside a new user account: How to test an issue in another user account on your Mac. This can let you know if the issue is isolated to your user or system-wide.
Let me know how it goes.
Have a fantastic day!