The brake cam won't actually be stuck, it's just turning too much when you apply the brakes and getting stuck between the ends of the brake shoes. The brake lever on the back of the brake drum assembly is attached to the brake cam, so when it sticks and you pull backwards on that lever, you are unsticking it then. So when you pull the brake drum off, it will be in the position it's in, in the picture above.
I can already tell you, by your description of what it's doing, that the brake shoes need to be replaced. I don't know if it will have any mud and water inside or not, it depends on if it has been ridden in a lot of mud and water, but if it does have mud and water in it, just clean it out good. You already knew that anyway. LOL. I'm pretty sure the brake shoes being worn out is the problem. Remember, the brake shoes will probably still have a lot of pad left, but they'll still be worn too much. I think they designed them that way so you can't wear them all the way down to metal and destroy the brake drum.
As far as greasing, it's a good idea to unbolt the brake lever and pull the brake cam out and grease it. You might be able to get by without doing that, but if you don't do it, and the brakes still stick, then you will have to take the brake shoes back off and do it anyway. If you don't take the brake cam out and grease it, at least put a little grease on the flat sides of the cam where the brake shoes touch it. Just use a thin coat of grease, you don't want to get any grease on the brake shoes or they won't stop you.
Most 4-wheeler problems are caused by a loose nut connecting the handlebars and the seat!!
You only need two tools in life -- WD-40 and duct tape. If it doesn't move and should, use the WD-40. If it shouldn't move and does, use the duct tape.