Been trying to diagnose/fix this problem for a bit now, I've seen similar problems in searching the forum but not my exact problem. My rear derailleur (Ultegra) when I try to shift all the way to the lowest gear/biggest cog, the chain tries to move to the cog, but doesn't get more than about 1/4 of the way on and slips back to the next smaller cog, and as I pedal keeps doing this.

It's not an issue of cable tension, the derailleur appears to be on straight and not bent. Seemingly it would be a limit screw issue, but when I loosen the L limit screw, the derailleur doesn't move out at all, and when I try to manually push it out, even while loosening the L screw, it won't move out all the way under the big cog. Seems about 1/8-1/4" to the left of where it needs to be to be truly under the cog, and no matter how much I loosen the screw and try to push it out, it won't go all the way. Any ideas? Or might my derailleur just be faulty? (should add, I've never fallen onto the right side such that it would have been damaged).