Car shakes side to side

To add, if the shaking is felt in the steering wheel, it's probably something with the front two wheels/brakes or related parts. If the shaking is felt in the whole car, it is likely something with the back wheels/brakes/related parts.
 
Maybe not specific to a Town Car, but the only times I've experienced true side to side shaking (more like a side to side wobble), it ended up being a failing rear wheel bearing. If that's the case, it's best to replace both sides since the other one is likely to not be far behind.

At the very least, check all your lug nuts (don't ask how I know on that one, LOL).
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Were you bumping the humpty dance? If yes, I think the car was just getting down. If not, I'd vote rotor inspection. If you very very lightly apply the brakes at speed, do you feel a little feedback in the pedal?

Driller, oddly enough, in my life, I've had a wheel leave a car on the interstate, then a completely different vehicle 10 years later, rolled off turning into my driveway. Neither event was tied to any recent work either.
 
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