How do i change the differential in my 1990 Corolla?

1990 Corolla.

Was drained when I filled it up with 80W90 gear oil.

Auto transmission.

US calif. Model. 4AFE engine.

I think something gave way inside(probably a tooth on one of the gears). It starts rolling, I hear a loud crack, and the left wheel shakes.

It started doing it when it turned only, now it does it going straight too.

CV joints are new and OK.

Tie rods Ok(new)

Steering perfect.

Brakes perfect( new caliper) New pads.

Auto transmission: new filter and fluid. Seems to work smooth( Then again, I have never seen what a broken down auto transmission feels like; anyone knows the symptoms of a faulty transission?)

My diagnosis is that my problem is the differantial, but i do not even know if i can just replace the differential alone(it is front wheel drive) or if it is part of the auto transmission.

ps. I have both Chiltons AND Haynes repair manuals, but neither talk .... about the differential. They go off track talking about the all wheel drive wagon transfer case.How do i change the differential in my 1990 Corolla?
I'm a little skeptical, "a loud crack" doesn't quite do it for me. I would inspect the suspension and CV joints again.

However, the car is a front wheel drive so the differential should allow slippage as the wheels turn. Fast repetitive slippage and seizing could possibly cause the wheel to shake. 80W90 does sound a little light, I would have guessed 85 or even 90 weight oil do to the direct connection to the transmission. The differential should be bolted on to the bellhousing, so you should be able to remove it without the transmission but I wouldn't know for sure without looking at it. I suggest removing the differential cover and checking the gear teeth for wear or any metal loose pieces of metal (there should be a magnet at the bottom that collects these). A bad transmission wouldn't cause the symptoms your describing. Normally when an automatic transmission fails it tends to burn fluid, slip, shift really rough, sick in one gear, not go forward or not go into reverse, hell I've even seen cause the engine to die out. You might be able to find one in the junkyard.

Anyway, good luck with that, I hope you figure it out and get it resolved. I'm curious to know what it is.