This is not too unusual. Question 1: with the derailleur in the small cog, is the cable taught? if yes, then the backing out of the limit screw may not have any effect because the cable won't let it move.

If not, it might be helped by the "B-screw" on the back of the derailleur which will impact how the chain enguages the freewheel. This would be unusual. Your cable may also be corroded such that it is not sliding smoothly. Worst case, you simply need to back out the limit screw such that the derailleur appears to slightly overshift (towards the dropout).