Eljay,
I've had a hard time getting those bearing carriers off in the past, if the machine has no cab. Rust will bind the carrier in the bore of the transaxle housing. Lots of WD40 and driving screwdrivers in from all sides. Try and avoid the corners, it is cast iron. If it moves a little, then tap it back in and repeat the process. The shims can usually be repaired, if they are not rusted too badly, or replaced with the identical thickness. No one likes to talk about using a hammer and a screwdriver but we've all done it.
When oil soaks the brake linings they have to be replaced.
It's a good idea to reseal both sides, including the carrier o-rings and the 2 differential lock seals. If just doing one side, there is a possibility the other side will start leaking because you have disturbed it unless you lock the brake on.
With the brake assembly off on one side, lift the side gear shaft up and down, noting any play. There should not be any. Use short bolts to hold your carrier tight against the transaxle housing for this test. Phil