There are many things that can cause this.
Most of these have a belt that runs the hydro-static drive. It is possible that this belt has broken, slipped off one of the pulleys, or just wore out. It may need replaced.
The manual is right , these hydros are not meant to be serviced.... at least by the customer.
Some service centers can add fluid if needed, but this doesn't happen very often.
This model is notorious for having the hydro-static transmissions go bad, so this may also be the problem.
I would suggest checking the belt and see that it is good and make sure that it is tight as well. Sometimes the spring that goes on the idler pulley arm will break and cause the belt to get loose.
Also sometimes the pressure relief lever on the hydro can get stuck in the off position. Try moving this several times and see if it gets better.
Other than these things, you might be better off letting a good repair center take a look at this. It probably would save you money in the long run.
I was looking for a belt but it was dark and difficult to see. What is the easiest way to get access to the belt?
There is no real easy access to the belt.
But usually it is located just above the mower deck. You may have to remove the mower deck to get to it.
You have a 1000 series tractor.
I wasn't sure if the hydro-static trans had a belt so I didn't spend much time in the dark looking for a belt
There are several sections of the manual, and you may have to check them out to see which one you need.
You should be able to see the belt near the back of the transmission as it must connected to a pulley on the transmission.
Here MAY be the correct section of the manual for the drive belt.
If you need any more help with this, you can contact me here at this question.