It really does depend on your skills and tool you have.
It is easy for me to sit here and say I would pull the frame and re-install it correctly.
And is still the best advice.
But if you do not have a sawzall, air compressor, finish guns, etc ...
Maybe it would only open a can of worms. Maybe the door works fine as is and just concerned about the drywall cracking?
A professional drywaller will bridge the door frame with drywall and will prevent cracks from the edge of the door frame going up the side of the header. Will be no drywall seam in that area.
Your issue really sounds like a poor drywall job then a bad door .. or both
Would be better to make a post in drywall then carpentry for the cracking and proper ways to fix it.