I have the JD repair manual for the LX200 series tractors and it says that the correct transaxle oil is 10W30 synthetic. On your LX277 AWS the transaxle is a Kanzaki K62B (the rest of the LX200s have a Kanzaki K62A transaxle).
The transaxle capacity is 2.5L (2.7qt). Make sure your drain pan will handle that much.
There are two drain plugs located on the bottom of the transaxle. They are located near the axle shaft on either side. If you have trouble figuring out which ones they are, let me know. I tried to scan in the picture of them but the file turned out to be huge and I don't do a lot of image manipulation so while I know it can be made smaller I can't do it easily.
Once you get it drained and refilled, you'll have to bleed it. To do that, you need to put the tractor up on jackstands so the rear wheels are off the ground. Start the tractor and run at slow engine speed. Press the forward and reverse pedals until the wheels start to spin. According to the manual this can take 5-10 minutes. If I recall, it took longer than that when I did it once, but that was after having the transaxle rebuilt so it was *really* empty. I think that after a simple drain and refill it didn't take long at all.
Hope this helps...