Usually its not the control arm itself actually...at least very rarely.

Lift the car up and grab the control arm and try to shake it back and forth. There should be no movement ar at the most very little movement. If you can feel it rock the arm in its mount point then you may need new bushings which can be done at home fairly easily.

When they do the alignment they should check these components though so I'm not so sure really that wil be the problem.

Take the wheels in to a different and reputable tire shop and have them check the tires out 100% because I am wondering if it may be the tires themselves at this point. Possibly a flaw in them that is throwing the car off.

Hard to diagnose without being there but these are the best guesses I can give right now.