Generally, the whine you are describing is due to either improper backlash at installation, or, overworking the gears during break-in. When overworked, the hardened material suffers lubricity challenges & begins to wear away. It'll continue until something breaks. :-(
I'd pop the cover, flush the fluid, check backlash w/ a dial indicator & dust to verify contact pattern. If you've miled the gears up, there may be no good fix other than replacing the set, as adjusting a set of gears with a established wear pattern via improper setting will cause new 'noises' and/or fatigue risers.
It is good money spent to have a reputable shop set your gears if new to doing it yourself. Sometimes it can take four or more adjustments to get backlash & pinion depth set correctly & a pro can accomplish the task in short order for around $200.
Good luck -