More often than not, the problem you describe is caused by wear and tear in the control linkage. This "slop" in the various parts means that you do not have good control over the spool that diverts the oil to the drive motor.
While I agree that you should eventually drain out the milky oil and install new 20W50 motor oil, that won't solve the problem.
Everything in the travel control must be inspected and repaired.
- new top bushing in the dash
- new ball joint at the spool
- repair the hole in the plate at the bottom of the travel lever. It is probably egg shaped.
- replace the metal rod that goes in that hole, It is work and perhaps cut.
The spool moves about and inch in total but your hand moves more than 6 inches when operating the travel lever. In that inch of movement is forward, reverse and neutral. It does not take a lot of free play in the linkage to cause problems like the one you describe.
All of the above is part of the restoration process you intend to perform. The best time to do it is when you have dismantled the entire tractor.