Rip it out and start over. It is past it's retirement age.

What's so special about the cabinet that you want to keep it.

Nothing special but we have kids in college and can't replace them yet...

Sounds tricky, consult with a paint store on a low VOC product

I recently painted my half bath cabinets with the strongest primer available from Sherwin Williams. I had originally used it on the ceiling to cover 50 years of tobacco smoke, and it worked wonderfully. However I will say I did first wipe everything down with mineral spirits and TSP to help eliminate any smell. But on the cabinets, once the primer dried I used a regular latex paint over it. They were white and gold before and now they are a light beige.