Nah, it shouldn't do that. Sounds like they didn't get the alignment right. What type of lift did you do, coils, spacers ? If you put new coils in it and didn't give them a chance to settle before your alignment, the alignment is probably off again. If you just used coil spacers that wouldn't have any effect. It could also be your steering box is too tight/loose. Did they check that during the alignment? A good shop would, and they would also test drive before and after alignment. Sometimes with a lifted vehicle, you'll have to find an alignment shop that has experience with them to get it done correctly. It's not just a matter of getting the " specs" to stock, it's driveability, and only a tech with lifted vehicle experience can get it right.
It could also be wheel bearings, worn bushings in the control arms, tie rods worn, ball joints worn. Check all these areas for loose or worn parts.
I have near 4" of suspension lift, modified control arms, no sway bar, and 32x11.5 tires and mine drives steady and straight. It's all in the alignment. Ask around in your area to find a shop that has lifted vehicle experience.