Originally Posted by Big R
...then set the L screw by tightening the cable with my hand rather than the shifter...
The cable tension has absolutely nothing to do with the L screw (low limit adjustment) or the high limit for that matter. Those screws simply establish the limit of the derailer's overall travel and have nothing to do with your problem, except maybe your high limit was not adjusted properly before the cable was attached, which would present indexing issues.
Shift to the small (12) cog. Remove the cable from the derailer. Turn in your barrell adjusters. Adjust the H limit screw so the chain is lined up with the small cog, check for smooth operation by turning the cranks and making sure the chain doesn't catch on the next cog. Double check the shifter to make sure it's in the high/normal position, to match the derailer's position. Now re-attach the cable. Check the shifting and tweak the cable tension as needed with the barrel adjuster. Now shift to the largest cog, if it won't go then loosen the L limit screw a bit. Once the chain is on the large cog, tug the cable by hand and ensure that the limit is keeping the derailer from traveling past the cog and into the spokes, adjust the L limit as necessary.