Mike, the "binding" feel, and the fact you see it try to grunt 1/2" movement, tells me that your ram is working, and has sufficient power.
OK: regarding the 1/2" movement: did you do as I suggested and have a 2nd person operate it, while you crouch down low and observe? When you do that, make mental note of whether or not it's having a slight pull towards one way or another. Versus uniform 1/2" movement equal. If it tries to go catecorner even in the slightest, then here's what you do:
Rather than 2 come-alongs for equal pull back-wards, you want to only pull on the side which is resisting movement. Like to "straighten it back out", so-to-speak. And THEN do the both sides even pressure to pull it back.
If you're saying this problem repeats itself after 20 dumps, then that tells me it's more than just a season of dirt that got frozen in there via lack-of-use sitting. That tells me your dump track-slots must have a crimp or bend in them. The good news is, that there NO SHORTAGE of used 2TE3 and 2TE4 parts to choose from. Heck, I'll sell you some CHEAP off a few we're cannibalizing here. Just cover shipping and a token of time
But back to your problem:
If the effort to get the door to go back exactly straight is not the issue, and since as you say, the ram IS trying, then the come-along trick should have worked. You might also try bottle-jacks to push outwards on it. I don't have a 2te3 or 2te4 pix here now, but seem to recall that you can anchor the back end of a bottle jack on a stationary part of the frame around the gutter broom assembly arm area. And then exert outward pressure. Either on the site resisting movement, or equal, with bottle jacks on both sides. Again, once you get it started (beyond that 1/2"), then the ram should be able to finish opening it on its own.
Or yet another option: Drive the sweeper so that you're backed up close to an object to serve as an anchor. Like a light-pole with concrete base, etc... Get a cable or chain wrapped around the pole. Then the other end hooked to the perpendicular fins of the bottom of your hopper door. Then you merely drive forward (away from the pole), while operating the ram to open it.
As an utter last resort, I can only think of hand-shoveling out the entire hopper full of debri, to take all the weight off the load. Or at least 1/2 or 1/3 of the load, then try again, etc... But that seems extreme, since weak rams don't seem to be the issue here.