Worn sway bar links would generate a rattle when going over uneven pavement. But they would not be the source of shaking/vibration.
Shaking/vibration, if only when braking, would be signs a warped rotor or an out-of-round drum.
Shaking/vibration all the time, which may get worse at increasing road speeds, would be a sign of a tire tread or sidewall belt separation, or a bent wheel (if the car hit a curb or pot hole), or a balance weight on one of the wheels came off.
If your 2 older tires are still under warranty, and you suspect the problem could be coming from the tires, I would take the car back to the place you bought the tires and ask if the balance could be checked on the tires to see if something is wrong with one of the tires/wheels.