Maybe you need a safe cracker with sand papered fingertips? 😉
I was thinking was it was AM on the cart causing the amplitude to beat up and down.
If it is the deck itself, then the thing speeding up and slowing down would cause a frequency modulation, and that would also warble.
I would still suggest a short capture of a sound file as a way to see which one it is.
Then knowing the nature of the effect, we should be able to find the causal mechanism.
