Light Action shifters don't often fail, so I'd go there last.
Even then, there are both 105 and Light Action 6-sp sets out there.

I don't get his "go to DA shifters" solution. Upgrade to 10sp?
That's a lot for a shifter problem. Wheels, cassette, RD, chain...

RD hangar, wheel seated well and straight across, check the cable clamp point on the RD.
Also check the cable routing under the BB, it could be hung up or have missed one of the channels....

I've got a set, but they're in use. I'd go 7-sp SunRace if I could find a 7-sp freewheel....