Never assume that you can safely remove a battery from anything.
Now I am not talking about the AAA batteries in your Walkman. What I am talking about is little watch batteries on circuit boards, and little plastic battery packs that serve unknown purposes.
Sometimes there are suicide batteries, and if you take them out, then the device dies permanently. While many other batteries will be the only thing keeping your data or settings alive. Always check the documentation, if possible, before removing any batteries.
If you can't figure out if it is safe to remove the battery, then try wiring up a second one using a soldering iron, and then removing the first one. It might sound stupid, but it could possibly save your data, or even your device.
Your previously wet circuit boards might work fine for a while, only to fail later. Try carefully removing all the socketed chips and cleaning the legs with very fine sandpaper, and then reseating them. This might get you going again. Be very careful not to bend or damage any of the legs, as some of them are going to be fairly fragile.