John Deere D140 - Lost all transmission fluid, now no movement.

Hello, first post here, hoping for some insight. Here is the sequence of events:

First start up of the season (battery dead, don't think that is related), let idle for some time, then went off to inspect the yard. Got to a point where I needed to reverse, I could not get it to move at all in reverse. Mover could still engage and run. Decided to try and go forward, would not move forward either. Engine is running fine, mower can be engaged fine, just no movement either forward or reverse. Have to put it in neutral and push it back to the garage. I notice where I initially started is and let it idle has a puddle, not water, oily.

I now have it in my garage on stands, mower assembly removed. The belt to the transmission is in place (on all the pulleys) but is VERY loose (is it like that when engine is not running or something?). At the rear of the unit there is oily mess along the rear right hand side, left hand side is clean and dry.

I am wanting to diagnose and do as much as I can before I need to take it to a pro. I have read that these transmissions can be removed and have the fluid changed? I am hoping I can remove the transmission and fill it back up, am I being too optimistic. Thank you for any help.

Is the oil coming from the top below the fan then running down , there is a seal that can be replaced. or is it leaking around the bypass lever , there is packing under it