Ok here it is, 1987 Trek 560. The dropouts are semi-horizontal with the opening facing forward (bad design IMO).
So on the first ride the drive-side skewer slips, I almost go down. I thought it just wasn't tight enough.. go out for a second ride after tightening it down really hard. Slips again, almost go down again. So something is wrong. A friend suggested the rear axle is too wide, preventing the skewer to tighten down on the dropout but as you can see in the pic both end of the axle only cover 1/2 the dropout width so I don't think that's the problem. My second thought was the wheel is too far forward in the dropout so I scooted it back as far as it would go, thinking there would be fresh paint to grab in to. Didn't work. Third thing I tried is a new skewer cap, this one is wider. Didn't work. So the last thing I did was put some clear fingernail polish on the dropout, but I haven't been out to test that yet.
Got any bright ideas?? It sucks not being able to enjoy a ride because you can't stop thinking about crashing..

Current position:



axle 1/2 across dropout; also you can see the fresh steel from the slipping at the front of the dropout:



Original cap on right (smaller diameter but more surface area), current on the left (bigger diameter, less surface area):