Loose hinges will cause problems.

Your problem is not a loose hinge. The bottom hinge is loose because of the problem.

Which is a racked(out of square/plumb) door frame. Basically, while the door is still square/plumb, the frame has shifted and is a parallelogram instead of a rectangle. That is fairly normal, occurring as a house settles over time.

The quick and dirty fix is to shim hinges or remove the part of the door that is rubbing. Both are temporary fixes and will result in more problems at some point.

The correct fix is to return the frame to square/plumb. That is done by removing/cutting the screws/nails used to install the door frame and reinstalling it plumb and square.

Then you use longer screws to remount that loose hinge and the door is correct again.

What handymac said. Plus you need to assess the reason for the frame racking. A small amount of settling is normal. But, if there is a larger issue behind it, you need to know. Take a 4' level and check the floors, walls, etc. to see how much settling has happened and where.

If you can screw in the screws on the lower hinge at all, then it means that is the source of the wiggling, and needs to be fixed. However, making that hinge tight will not solve the sticking problem (in fact, it will make it slightly more noticeable). There is nothing wrong with some shimming in order to fix this if it's done properly...I've done it successfully several times. What you will need are thin, uniform shims: rectangles the size of the hinge rabbet (cut some cardboard from a cereal box and make yourself several pieces with holes where the screws will pass through). Take the screws out of the side of the hinge mounted on the jamb, slide in one piece of cardboard, and reinstall the screws. If this works, fine. If it still rubs, install a second piece, etc.

One other issue is whether the wiggly hinge may have stressed the screw holes; if so, the screws may not stay tight over time. If this occurs, re-post here for tips on how to fix this.

Lastly, most door hinges don't have a hinge spring. But, even if yours does, the contractor's advice does not sound wise, particularly if you don't have much knowledge of how this stuff all relates. It sounds like a recipe for you to damage the hinge and make the problem worse.

Try the steps above and then report back, please.

Either shim the lower hinge outward, or deepen the mortise of the upper hinge. Either way has the effect of lifting the lower knob side corner of the door up off the threshold. Don't overcompensate to make the top corner rub the top of the frame.
Hinges do wear, and will eventually wear out. At the midpoint of their service life, they can be adjusted. Shimming the lower hinge is better, because no material is removed at the jamb, but sometimes both must be attempted at once to correct the hang.
Casey

Hi, Look at the top revel between the door and jamb. The gap is smaller on the hinge side and bigger on the latch side. At the top hinge, remove the screws closest to center of the jamb and insall longer screws about 2 or 3 inches. You want to catch the stud behind the jamb. You want the screws to pull the jamb a bit. Now shim the bottom hinge if needed,
Good luck Woodbutcher

What "woodbutcher" said. Look at the door. If the gap is larger at top/ strike side, make sure top hinge is screwed in firmly and mortise deeper if needed. Then shim lower hing if needed.