We replaced the front door with a same size oak prehung door. The jam is level at the top and sides but door is hanging down at the handle side about 3/8" and it doesn't close without pushing it closed unless we lift the door up by the handle. When we replaced the door we replaced one of the screws in each hinge with a 3" screw so that it would have extra support. The door is tight and even on the hinge side. What is wrong and how do we fix it?
Top hinges not set correctly.
Jamb is not square/plumb
Floor might not be level (an often oversight is not checking - and correcting - an out-of-level subfloor)
Because you have to lift up on the door to latch it, it sounds like the frame is out of square, or the floor is out of level and the frame plumb.
At any rate, you could play with the hinges until the Iceman Cometh, but your best bet is to remove the casings and figure out what's wrong from there.
FWIW, when I set doors - interior or exterior - I don't use levels or plumb bobs any more. I set up a PLS180 self-leveling cross-line level. Using this I can determine if the floor is level or not and by how much, and shine the verticle beam on the jamb edge when installing the jamb, so it is dead on every time.
Good luck. Doors are great. Unless they aren't working right.