1998 Mercury Marquis shaking

Try rotating the tires front to rear and see if it changes it, if so you know it is the tires. An Alignment wont usually make the wheel saw back and forth like that, incorrect alignment causes pulling usually. Does it do it all of the time or just when you are on the brakes? If when braking, you haev warped rotors adn they need to remachined/replaced depending on their condition

So you have replaced 2 of the tires and it still does it. Even if they are on the front? If so something is wrong with the car. If a tire is bad enough to make a car shake, you can usually see it by spinning it

I fht efront end has something bad enough to make it shake, you would think the alignment guy would have caught it, have another shop look at it, or lift and support front end an dsee if you can find anything loose