Well, if it isn't one thing, it's another. After celebrating getting the A3S/A743 rotating again, I've found a problem. The SWR shifts I mentioned? On a dark night, I went outside and looked up at the antenna while WSJT-X was transmitting on FT8 trying to contact someone. I tried this on 40 and 20m.
I observed one of the 20m traps on the driven element light up with arcing using 100 watts. It was visible in the drain holes and end caps of the trap.
At the very least, I'll need to take the driven element down and rebuild it.
Update: I've tried this experiment on 15m, and I see no SWR shifts even using 500 watts. I think the problem may be confined to the 20m trap(s).