Shaky rear end

You need to get the car on a hoist and fully inspect the rear end for damage and worn suspension bushes. Make sure none of the hardware has failed and it has dropped a sway bar link or any other suspension component is compromised.

I do not understand the last part, but for a shaking or shimmy in the rear, the best suspect is a tire with a broken belt. You may be able to feel the bulge in the tread, but be very careful. Broken belts are often accompanied by wires sticking out of the rubber. Those really hurt when they poke your hand.

You may need to raise the tires off the ground, then spin them by hand to look for the hump in the tread. If you do not see that, there is one more thing to look for, but you are better off having someone at a tire and alignment shop inspect them.