base timing keeps advancing by itself...

If anything was slipping slightly, wouldn't that retard the ignition? Not advance it? It sounds like it's adding 10degrees pretty frequently. Has it made a full circle?

Something odd is going on. If you shut off and let it cool overnight, does the timing stay advanced? Or does it go back to normal? Electronics can do strange things when hot. That would be my first suspect.

Have you verified the ignition module pickup wiring isn't backwards? It's easy to do, and these days there are units that come out of the box incorrect (I haven't seen it, but reputable people have posted about it). From memory, whichever orientation causes the least advanced condition is the right one.

Last thought: I know OP is sure it's actually gaining advance. But I'd expect similar behavior if the idle mix is lean and the idle hangs a bit after a drive. Higher rpm makes for more advance.. I would try something. The next time it does this, don't move the distributor, instead richen the idle mix a tad and see what happens. Or even drop the idle with the screw and see if the advance goes back to what it was set at. I have an engine that occasionally hangs before the idle drops back to normal. I sometimes just slip the clutch a bit to lug it down and then it'll stay there. Ever since I readjusted the electric fan switch to come on sooner it, it doesn't do it.
Correct. As the timing chain stretches, timing is retarded. I think looking at the timing chain here is a waste of time. I would try another distributor, as I believe that's where the problem lies.