Do not attempt to make alterations to the door itself. That will not work.
I am a pro handyman taking care of a homeless shelter. Our doors get lots of abuse, and some of the frames are kind of old. I get doors sagging from several different causes. Most of the problems can be corrected.
Try lifting at the swing edge of the door. Lift straight up. Try pushing straight in toward the hinges from the swing edge - pushing the door straight, through its width, toward the hinges. Did anything move? Were you able to shift the door back toward its frame? If so, excellent - you have a chance at some easy home fixes.
First check the screws in the hinges are socked all the way down. If any of them are loose, including the ones in the bottom hinge, the hinges can droop out of position and allow the door to sag. BUT don't overtorque them trying to sink them. If they're tight, they're tight. Don't strip the screw holes trying to make them tighter. If they aren't tight, carefully sock them down. If you have hinge screws going into wood that won't tighten, be prepared to completely remove the door, remove that top hinge, and just glue in some hardwood splinters (a small bundle of toothpicks works fine, tapped home with a hammer) into those holes. Give the glue some time to dry and trim the splinters flush with the surface, then you can drive your screws back in and they should hold great.
If the hinges screws are tight, shut the door and look carefully at the alignment on the hinge side. Does it look tight at the bottom, and more open at the top? If your screws are tight but that hinge gap goes from tight at the bottom to open at the top, your hinges have fatigued.
You can replace the hinges, that's easy. But you can also try:
With the door shut, pop the hinge pin out.
Using a Vise-Grip, grab each of the hinge barrels and bend it, carefully, just a little bit. If you look carefully, you can see which of the hinge barrels are connected to which leaf of the hinge: bend the barrels on the frame back toward the wall, the barrels on the door back toward the door. You're bending them away from each other.
DON'T BEND FAR. Seriously, just like 1/16" is plenty - between these going this way and those going that way, that's 1/8" total separation, and that's a lot in a door gap.
Now. Lift the swing edge of the door and while holding it up, drop the hinge pin back into place.
If the frame or door won't bear up against you bending the hinge, then completely remove the hinge - leave the door open for this, and supported with a wedge under the swing edge - and bend the hinge barrels with the hinge leaves clamped in a vise. Just keep track of which leaf is which and bend the barrels accordingly.
What this does, when you drop the hinge pin back in, is force those leaves close together when the door is shut. It won't much affect how the door opens, but it will make a big difference how it fits in the frame.
I check back on my answers from time to time, if you have any further questions add them to your question as Additional Information, and I'll help if I can.
Good luck with it.