I have a 2003 suburban 2500. it started last winter and has gotten worse to the point now where its not working at all. it all started last winter when I tried to clear snow off the back glass after work. during the winter it is garage kept. when I went to clear the snow it didn't want to move so I thought it was just froze to the hatch. I turned it off and went and took the snow off by hand and freed the arm from its frozen position. once it was free it moved very slow across the window. by the time spring came around it was so slow you had to help it across the glass. then as time went by it would lift off the hatch and just stop on the glass and go no further. I was able to get it back down with some help. now it won't move. when I move the switch inside you hear it hum like its trying to work but doesn't. I've read about the bad ground melting the harness but don't know if that's the cause or if the motor went out. its not a big issue but it would be nice for it to work again especially when it rains.

 

I had the same problem a couple of years ago. Mine was the motor was getting weak. Since you think that you can hear the motor trying to run, I would think that the wiring is OK. I would remove the wiper arm off the motor to see if the motor turns good. Taking off the arm is no problem, just remove the cover and it's one nut holding it on. The post that the arm is bolted on too is splinded, so make sure you lift straight off so you don't ruin the threads or you have to buy a new motor. A battery connection puller works great with no damage possible. I would use one because the arm is probably on there pretty good if you have never had it off, I think that they are fairly cheap at a parts store. Once it's off check the splines, maybe they're stripped off and that's why the arm won't move. Once you get it off see how it works, turn the wipers on and try to hold the spline putting some tension on and see if it still moves. Mine worked with the arm off but as soo as I put a little tension on it, it felt like it almost froze up.
When it snows, if you use the wipers to remove the snow instead of removing the snow by hand it really wears down the motor.
Hope this helps