DCB0

There is no reason to adjust your limit screws. This is the most common and, frankly, incomprehenisible mistake... they are limit screws and the problems people try to solve by adjusting them never have anything to do with limits. I probably made the mistake when I was starting out, too, but it is still weird how often the mistake is made.

Anyhoo, yeah, if your chain is worn past a certain point, lets call it 'point NFG,' then the cassette is also worn to match and a new chain will not mesh at all with the new cassette. Get a new cassette and the problem will go away...

but since you fiddled with your limit screws you should probably start anew with derailleur adjustments, and, once set, do not touch the limit screws again until you have a problem with the limits of the derailleur movement.

So install a new cassette, go to ParkTool.com, find the instructions on derailleur adjustment, and follow them step-by-step and don't leave anything out.

Good luck!

Edit:

I just re-read you post and I think you might have been talking about the B-tension screw on the back of the derailleur, not the limit screws... well in that case forget what I said about re-setting your derailleur. But still look for info on how to set the b-tension screw, as it is unlikely that 'all the way in' is the ideal setup.