I can mostly visualise what you're trying to do but some images of the setup and so would be very helpful.
As I understand it: this is a cutting wire, that forms a loop around two rollers, and then is moved back and forth for a small fraction of the length, the speed and how much movement are set by potentiometers.
The only part that I don't fully get is why you use a pot to set the distance moved when moving to fresh wire, rather than the time the "fresh wire" button is pressed, as it's going to be hard to know how to set the pot when moving the wire.
Adding a display (simple text LCD) is easy.