Well, I may have solved my own problem, but I will post my results here as I go along in case it helps someone else in the future.
In the lower left hand side of the control board there are two TC4011BP NAND gate IC's.
Datasheet:
The circle on the left reads a much cleaner square wave than the circle on the right. Both have similar input signals coming into their logic pins.
This leads me to believe that either the chip is bad, or that perhaps the signal that is going into either pin 1 or 2 is slightly off on the IC on the right. I am tempted to take the two IC's out and swap them to see if it exhibits the same behavior with a different IC, but I fear of breaking the pins on an old IC. I'm going to try next to get a closer look at the input signals on pins 1 and 2.
Update:
Swapping the IC's didn't help, so apparently the problem lies before that point. Tomorrow I need to trace around, take this thing further apart, and find out what path the signals take to get there.
Update 2:
Curious.
On the back of the board that contains the two NAND gates that produce the square wave I found an oddity.

There appears to be a filter cap placed over two of the pins that control one of the nand gates on the IC that gives a cleaner signal... which admittedly makes sense and solves the mystery as to why one is different. Judging by the traces all the band hates that I saw signal on share pins except the one that is messed up looking. Weird. Why two signals? Why two chips with multiple nand gates?
Now the question is why there isn't one on the other NAND gate? Does anyone know? Should I add one?
Here is that board in full:

Update 3:
Problem not solved, but when I put it all back together it was producing a cleaner waveform. I’m guessing lots of these connections were in poor condition and needed reseated.