If that is a hydro-static transaxle, they are usually not designed for owner maintenance. The makers figure if you like the tractor, you will invest in a new unit, and forget any maintenance on the old one! Although, there are a few folks here that might have a fix for the problem.

Oil is 80 W 90 gear oil.

Here is a link that might be useful: IPL for 917.271013

If you are noting dampness on the outside of the transaxle as opposed to actual dripping, don't sweat it. If there is leaking, pull the battery, set the park brake and pull the driven pulley. Once cleaned- you will see a small plug on top. let the oil container set in the hot sun and do your best to put some oil in the thing. You'll soon know what I mean.
The other version is much more involved which includes pulling the transaxle, cleaning, splitting, cleaning, swearing, cleaning, repairing the leak if possible, more swearing and cleaning- possibly in the end packing it w/ lithium and hoping for the best.