Not a transmission person, but depending on the brand/model of the hydro, it "might" be greatly improved by a fluid/filter change?? Make sure the drive belt isn't slipping.

You have the Tuff Torq K58 trans, it's a K46 with a charge pump addition. Life of a K46 is about 900 hours, slightly better for a K58 I would assume. Heat is the main destroyer of transaxles. Keeping the transaxle fins clean is a must to have it last. See the link attached for how to rebuild a K46, but it is very long (72 pages). Change the fluid may help and is cheap - use 5W50 synthetic motor oil - 2 Qts. Some have had great results just sanding out scratches on the internal parts. Here is a link that might be useful: K46 rebuild

Oh lordy, here we go yet again. Use the forum search facility at the bottom of the forum opening page and do a search for subject like "Hydrostat oil change, Hydrostat fluid change, Hydrostat transmission slipping. This is a perennial complaint and YES, the oil can be changed (no matter what the book or the dealer say). and YES, old, degraded oil will cause symptoms like this. Many are the hydrostatic transmissions still going strong that would otherwise have gone to the scrap yard but for an oil change, which is not that hard to do (you said you are handy, or know someone who is). :^)

There are a few possibilities, all having to do with excessive wear or loose fit in the directional control linkage. My first suspect would be the control arm on the transaxle case where the directional control linkage ends at the transaxle. The control arm attaches to the directional control shaft (aka swash plate shaft) with a bolt. The control arm has a square shaped hole and the control shaft has a square shaped profile where the two fit together. It is possible that the bolt holding the control arm to the control shaft has been loose and worn away the profile of the two mating parts to the point that the control arm and the control shaft are no longer locked together, but instead are now able to allow the control arm to move (when you move the directional control lever on fender) without affecting the same movement to the control shaft. This is going to require a close up visual inspection of the control arm where it attaches to the control shaft while some body operates the forward/reverse lever to determine if there is a lot of slop between the two components. Normally, there should be no lost motion between the two parts. You may have to remove the bolt that holds the control arm to the control shaft to see if the square hole and shoulder are no longer "square" because of wear.