The sawdust and banana "trick" has been around decades.. It's something used car lots did to quiet howling rearends. The theory is the sawdust particles and banana fibers form a cushion between the gears.. It sometimes works but most time it doesn't.. At best, it's a short term fix but not something I'd ever recommend..
The howling and/or whining the OP is experiencing is probably due to wear.. If the differential was recently overhauled and "set up" improperly, correcting pinion depth and gear backlash might eliminate the noise.. More than likely, the gear pattern has "taken a set" and the noise can't be eliminated without a major overhaul.. Personally, if the whine isn't very loud I wouldn't worry about it...
In my younger days I had a '55 Chevrolet that I did some drag racing with.. The differential was a 5:13 ratio.. The ring and pinion weren't a matched set but had a decent gear pattern.. The thing howled like a banshee during mating season.. Never gave me a bit of trouble..