Figured it out.
Though nothing on the interwebs told me so, turns out there are five positions on my shifters (105 STIs). I had the cable too tight (and probably the high stop screweed too far in). I'm still dialing it in, but now I have 1 on the small chainring (resting on the stop and the cable pretty loose), 2 is a small click in (mostly taking the slack out of the cable and pushing the derailer a little way), 3 slightly inside the middle chainring, 4 slightly outside the middle chainring, and 5 sitting on the large chainring. Seems I only have about 3 gears in the rear cassette when in the large front chainring before it starts rubbing.
I believe 2 is actually some sort of trim for the small chainring and that the shifter is "supposed" to skip it when shifting from 3 to 1.