I worked in a shop last year that was selling Specialized. We ran into that problem all the time. It basically comes down to the Specialized crank, or at least that's the conclusion we came to. We tried a new/better chain, which helped, but did not fix it. We then put a Shimano crank on the bike, setup the FD, and what do ya know, it worked fine. The spacing between the rings on the Specialized crank might be off just a hair, I'm not quite sure.

We eventually got it to shift sort-of-ok with the Specialized crank. I would suggest lowering the derailleur so that the cage is just barely clearing the big ring. Then, set the limit screw tolerances on the low side really close to the chain. I usually tighten it down until the cage rubs the chain (low-low gear front and back) and then loosen it up a little at a time until it just stops rubbing. Then, I make sure the cable is tight - I'll loosen the anchor bolt, pull the cable tight, then tighten the anchor bolt. Next, use the barrel adjusters to tighten the cable a little more. Once again, tighten until it rubs the chain, then back it off a bit. That's about it for the low side. The high side isn't as critical on these cranks, so I think you'll be ok there.

Also, you might have to play with the angle of the cage in relation to the plane of the chain rings. Sometimes I'd angle it to the outside to assist the chain going up to the middle ring. Also, that would help if the chain is falling off into the frame. Obviously, don't angle/toe it very much - I usually never tow it more than 1mm or maybe 2, but that's only in very rare cases.

Rev.Chuck has a good point - I've seen a lot of of the pivot bolts in ShimaNO front derailleurs loosen up over time. I am no in the habit of checking them if there's any problems or slop in any front derailleurs that I work on.

Good luck.. keep working with it. You'll get it. So you're a car guy, too? I am getting into it a bit myself.. Check out my toy/project/#1 money pit in my signature. It's a blast!