You don't actually aim to get the doors to hang plumb. The main objective is to get the doors to fit the opening, into which they close. Ideally this opening is plumb on both sides, sitting above a perfectly level floor, with the top of the frame being perfectly square and level. If this is the case, once you have fitted the door, it will be plumb, if the opening is not plumb, c'est la vie.
Having another look at the description you have given above, I suspect that your doors may be "hinge bound". This gives a characteristic springy feel to the door, as if you are levering against the will of the hinges. Basically it means that the hinge edge of your door is meeting up with the door frame, before the handle side of the door has reached it's resting point. There are several possible causes for this:
If the screws used to hold the hinges onto the door (or frame) haven't been screwed all the way home, or they have the wrong heads for the hinges, they may prevent the hinges closing all the way. This may be overcome by replacing the existing screws with slightly longer ones. (if they have worked loose, simply tightening them is unlikely to prevent the problem recuring).
If the hinges have been recessed too far into the edge of the door or, (more likely in your case), it's frame, this will cause the edge of the door to be butting upto the inner edge of the frame, before the door is completely closed. There are various ways you can go about rectifying this problem. Quite often it is caused simply by years of paint that have built up on the frame. This, in turn, increases the effective depth of the hinge recess. Taking the paintwork back to bare wood and priming and repainting, can often sort the problem out.
If, on the other hand, your doors are catching elseware, you simply need to close the door, while inspecting carefully from outside, and marking where it is fouling the frame. You then just plane down the door along the edges, where the problem occurs.