Honestly I would say the leak on one side has everything to do with it having issues as this is why it didn't have any fluids inside it when you attempted to drain it. This lead to the diff itself being damaged. I seriously doubt any kind of quick fix is available to you at this point through "thicker" or any other type of varied lubrications.
The damage is done so your options are, A. Can I live with the whines and or grumbles. Or B. rebuild it properly through a locally respected shop near you.
Unfortunately your hand may be forced to B depending upon the seriousness of the damage and it's affect on the diff.
Keep one thing in mind though. When a diff goes it can be BAD, real bad. Like suddenly send you sideways in an unplanned right or left turn into something solid or off of something depending on where you may be at that particular un-choose-able moment in time. Or it may just go.......ker-snap and suddenly all forward momentum is gone.
Your diff IMO, can be seen as a life-supporting mission critical part of your car with no less of importance then your brakes or suspension arm. Something in other words you don't want surprising you with the ker-snap as it were,
Just my thoughts on this. A diff going bad isn't something that should be ignored until your forced to deal with it.
