Is there a way to check the fluid level? I don't own anything with a hydrostatic transmission, unfortunately. But from what I've read in passing about the ones used in lawn tractors, it sounds like low fluid levels can lead to it slipping.
"Forcing" them also isn't good for them. Such as moving the pedal/lever further when your tractor's ground speed starts to slow down while going up a hill, while the engine speed remains the same. Unlike in a car, pushing the pedal further on a hydrostatic is simply trying to alter it to a higher gear ratio (faster ground speed), which is the opposite of what you need if it's starting to slip. In that case, you want to ease off the pedal and let it climb at its own pace, in a lower "gear", which is easier on the transmission.
I presume a similar scenario for a snowblower would be getting to a big snowbank, seeing the transmission slow down. and moving the lever further to try and speed the machine up again.
On at least some of the lawn tractor hydrostatic transmissions, it sounds as though they are often not rebuildable. So if they have problems, you can try to top off or maybe change the fluid, but beyond that, people apparently sometimes have to look at replacing them. I hope that's not the case here, of course.
Does the machine have a feature to disengage the transmission, to make it easier to roll the machine when it's not under power? I was trying to help a neighbor with her hydrostatic lawn tractor last year, which wouldn't move, and had spent 5-10 minutes checking over the "obvious" stuff, before I noticed a lever on the back, hidden by the bagger. It was to disengage the tranny. Flipped that back into position, and off it went