Hi Tim: The problem you are having probably has little to do with the doors, but is caused by sagging floor joists on the second floor. This will allow the door framing to drop at one corner, causing the door to rub the top of the door jam or drag on the floor/carpet. Unfortunately, the sagging cannot really be fixed short of jacking the center of the house up and putting in new supports. I've been involved with this kind of process. It is very labor and equipment intensive, and not suited to most homeowners.
Typical fixes involve cutting off the top or bottom of the door with a power saw to relieve rubbing. The best way to fix this is to take the door casings off, remove the door, cut off the nails holding the jam in place with your recip saw and reinstall the door jam by reshimming and plumbing within the door frame. This is a bit of work, but will give you the best looking results.