I suspect something has come undone in the front end. Does it make any noises while this is happening? Put your front wheels straight ahead according to the steering wheel position. Eyeball your front tire to your rear tire on the same sides. Are they more or less straight or is one or both pointing out or in a lot? Is the steering wheel straight when going straight? I've worked as a front end mechanic for a couple of years and I"ve never come across what you describe. If you had a recirculating ball steering gear I"d say it was way out of adjustment, but you have a rack and pinion when doesn't need adjusting. Right off hand without looking at it, I'd say the rack and pinion has come loose from the body of the car. Another thing it might be is a tie rod or ball joint went out on one side. If you want to, you can jack the car up, grab the tire at 3 o'clock and 9 o'clock and move it side to side. Is there any play. Then put your hands at 12 o'clock and 6 o'clock and try to move the wheel back and forth. Are you getting any play. With the car on the ground push on the front up and down. Are you hearing any noises? This is a real tough one because the problem could be just about any front end part that has come loose. Don't drive it anymore than you have to. Dangerous situation!
What I would do to make sure I wasn't being bilked is to go to several places where they do alignments and have them check the front end parts for wear and get an estimate from at least 3 places. Then when you know what is wrong you can get on the phone and check around for the least expensive repair.
It is difficult to estimate how much it will cost without knowing what is broken or needs aligned. btw all reputable shops will inspect the front end parts before they try to align it. You can't align a front end that has parts worn out. Then they will let you know what and how much. This is standard practice . If they try to tell you what it is without putting it up in the air and checking the parts go somewhere else. Wish I was there. Dave