I had a very similar problem with our 1997 AYP/Poulan 18.5 hp garden tractor. It's steep here, so I use heavy wheel weights and tire chains, which obviously adds load especially uphill, and mine was slowing to a crawl. I use our tractor for heavy cart towing and cut in heavily weeded/woody areas too, which tends to be hard on the drive and other belts. Mine have shortened lifespan from being cut at least as as often as they actually wear out from wearing through their fabric coatings on the drive surfaces of the belts.
The USUAL reason why I tend to lose drive speed while on uphill stretches is from air being in the system, so I've gotten in the habit of doing the driving several feet forward and immediately reversing for several feet, and repeating this cycle several times while on a LEVEL surface to purge air from the hydrodrive before even heading out to mow. (And sometimes do it during mowing sessions too-when I find a level enough place.) The mower repair place that did the warranty work the first 2 years told me how to do this.
I'm also used to worn belt-slippage issues, and deeply worn pulley grooves making the belt ride deeper in the "V" of the pullys. I've replaced probably every pulley on this machine over the years-some more than once-for wear, and too many belts to count. Both of these result in the ground-drive belt being looser than originally installed. So these were the next 2 things that I checked this time. Both issues checked out ok.
I checked the belt itself for oil being on it too-which can duplicate the same symptoms. No problem there either.
Read up on the model Hydro-gear transmission that mine has (314-3000) found the capacity and type of oil recommended (121.6 oz. 20W50) and then tried unsuccessfully to find the fill plug to check the oil level. (It's not where the exploded diagram shows that it should be on ours.) I'd figured that I'd just check the level to see if maybe it was a bit low on oil-after all this unit is 15 years old and has never been apart or even drained. If the level checked out ok, I'd figured that then maybe I'd try checking the filter to see if it was partially plugged up.
Spent over an hour air blasting and scrubbing dirt from nooks and crannies and searching the unit to find the fill level plug, but access is limited due to the Poulan having a ground engagement implement attachment hitch system and the large additional plate reinforcements, springs, and other hardware that carry the weight and attach to theoriginal back section of the tractor. Never did find one, so I tried checking the level through the larger diameter top vent hose connection and wasn't able to get into the trans far enough to hit oil. But I think that I got in far enough to get to almost the bottom of the top expansion chamber. Hmmm.
I figured that (if it was actually low) a little additional oil MIGHT make the difference between it foaming the oil when the tractor was in certain positions and that if it was already full, that the worst that could happen is that some would start coming out the top of the unit when it got hot enouhh and expanded fully. I started by adding 2 oz. of 20W50 oil through the vent, (less than 2% of the specified oil volume) reattached the hose, just to see if I could notice any performance difference. Crossed my fingers and went out on the steepest slopes that I normally mow. It made a BIG difference-and this was without changing anything else.