There is no other reservoir, you just take the back cover of the tranny off to drain the hydraulic fluid (actually Cub Cadet Hytran), then install a new gasket, and re-fill until it runs out the plug that you pulled off to feel the fluid. There is also a spin-on oil filter on the front of the tranny that you should change, again, a Cub Cadet part.
If this is something that just started, it is likely the roll pin in the drive shaft that has broken. Yes, your tractor is shaft driven, with a steel drive shaft that runs between the engine flywheel and the input gear on the transmission. Below is the same transmission, out of a Cub Cadet 125. If yours is a 1968, it is either a 105 or 125 (10hp or 12hp) but they used the exact same tranny.
This gives you a good shot of the oil filter, and the rubber, flexible coupling above it is where the drive shaft attaches. There, the drive shaft has a T which bolts to the rubber coupling. On the other end of the drive shaft there is a solid roll pin that fits in a notch on the flywheel. That is likely what is broken off and allowing the drive shaft to remain still while the engine is turning.
Now that I've looked at the picture, there is another option. On the very top of the hydro pump (silver thing) there are two big hex nuts with buttons in the middle. Those are the hydraulic dump valves. If they have corroded and stuck down, the transmission will act like it is not working. In reality, those buttons open valves that just let the hydraulic fluid through without doing any mechanical work (moving the tractor). During normal operation they should pop up. A quck test for this is to see if you can push/roll the tractor while the engine is off. If so, the dump valves are stuck down, and allowing the fluid to move freely. If you cannot push the tractor around, then this is not the problem.