The suspension gurus here probably know tons more than I do, but I'd guess the problem is related to the car being lower in the front than the rear. Normally, front struts are angled forward, like this: < / the arrow indicates the front of the car, the slash shows which way the struts are angled. This angle is important. If you draw a line through the strut to the ground, you'll find that this line intersects the ground forward of the center of the tire contact patch; this gives a stable and self-correcting steering system.
With the nose angled down, your struts will be more nearly vertical. This moves the center of the contact patch under or in front of the point where the line through the strut intersects the ground. This destabilizes the steering system and could be the cause of the car's wandering.
Another possibility is related to the rear suspension. Semi-trailing arms change toe as they move up and down. Being angled all the time, their toe changes may have more effect and cause the rear to wander around a bit.
Either way, the answer may be to make the car level again.
Or I could be full of **it!