Will it pull in reverse after the forward drops out? The first thing we always do is purge the transaxle again and re-check the fluid level. When you purged it the first time, did it need more fluid after purging?
The most common cause of this would be fluid failure after it heats up. The fluid boils and hydraulic pressure is lost. You could be losing the piston seal in the pump or motor due to distortion or galling, but the fluid would be more likely.
I don't see a reservoir or external filter set-up for the RZT50. You don't have one do you? Also, what's the model number of the mower? Is the terrain this is used on very hilly?
We use Castrol and have great success with it. You have the Kawasaki engine? That was a good choice.
I'd try purging once again and rechecking the level. Was the old fluid burnt or discolored?
The only place the fluid is pressurized is between the pump and motor, the transaxle itself isn't sealed air tight.
But it shouldn't have been white. Are you sure it wasn't water mixed in?
It has to be disassembled. You really need the Hydro-Gear manual for the transaxle to do this and they don't let them out to the public as far as I know.
The fact that the oil is foaming is what's causing your problem with the pulling. Now the hard part is finding why it's foaming. The only thing I can think of that's likely to cause that would be a problem in the center section, either the pistons or the section itself having a leak.
After looking at the schematic again, it might be possible for the bypass valve to be causing this. This valve is the one which you engage and disengage to freewheel the mower. If it's working loose, it could cause the oil to aerate, and you would also lose power. The valve itself requires disassembly of the case to access.
You have to have the service manual for teardown and for checking all the clearances. You can contact Hydro-Gear and see if they will provide one. You'll need the model and serial number from the transaxle itself before contacting them. Hydro-Gear doesn't allow it's Service Centers to give out this sort of thing and their website with the manuals is password XXXXX anyway so I can't give you a link to it.
To contact Hydro-Gear you can go to their website at http://www.hydrogear.com/
They're not terribly complicated. If you can rebuild an engine you can rebuild a hydrostat. They are a bit more clearance critical than an engine though.