Your shingles that are overhanging your gutter should be on top of a piece of metal flashing. If they’re hanging over your gutters by themselves, the problem can be is that through its property of cohesion and adhesion is it will adhere to the bottom of the shingle and wick back up into your roof deck. The deck is the plywood that the shingles are on. This causes rot at that joint.
It’s a very popular problem. It happens to many, many roofs. The solution to this is to install the drip edge flashing that should have been there in the beginning.
Very rarely should someone attempt to cut back shingles that are overhanging. That’s usually not going to solve the problem whereas flashing almost always will. Adding flashing after the fact is possible although it’s costly.