2002 Toyota Corolla Vibration When Accelerating

I have a Toyota Corolla CE 2002 (automatic) with 85k miles. It vibrates and makes some noise when:
1. I make a right turn and accelerate simultaneously.
2. When I have 2 or more people in the car and I accelerate (regardless of speed).

It kind of feels like the wheel is hitting' the wheel well and causing it to sputter and vibrate when making a left but it does not explain that when 2 or more people are in the car.

It also kind of feels like the car is forcing itself at a higher gear (ex. Trying to accelerate a vehicle, petal to the metal at 10 mph in 5th gear). I hit the gas to force it to a lower gear but the vibration is still present.

The "last" sensation it feels like is that the car is slipping from peeling out but I know that is not the case. It almost feels like a disconnect between the wheels and the engine/transmission.

I would have to say that I have experienced this more recently in the rainy cold weather.

Is this the transmission? Could this be the CV joint?

Any help would greatly be appreciated.