Your observation that the steering wheel shifts position tells us something is changing related to the alignment. That can be caused by a worn steering part, mainly tie rod ends, but more likely something else that's allowing a wheel to tip one way, and then the moving spindle is moving the steering linkage. The more-common suspects would be worn ball joints, worn control arm bushings, and a worn strut where the shaft can wobble back and forth where it exits at the top of the strut body. The best thing is to have the steering and suspension systems inspected at a tire and alignment shop. The people there are experts at finding the causes of noises, vibrations, handling problems, and bad tire wear patterns. Many do the inspections for free, especially if you have the repairs done there.
Here's links to some related articles that will show the types of things we look for and the problems they can cause:
https://www.2carpros.com/articles/steering-wheel-shakes-when-accelerating-or-braking
https://www.2carpros.com/articles/popping-noise
https://www.2carpros.com/articles/how-car-tires-work
Given the age of the vehicle, sagged coil springs are another concern. Conscientious alignment specialists will measure front and rear ride height, and they'll want to correct it with new springs before they'll take your money for an alignment. If a front strut needs to be replaced, they've been available for a number of years as a "quick strut" with a new spring and all new hardware already assembled and ready to install. That removes the huge safety concern when compressing coil springs to reuse them on a new strut, and it saves a lot of time. What costs you extra for the parts will save you in labor cost and give a much better result. Struts can be replaced on most vehicles this way in less than a half hour.
Less-common is a worn upper strut mount that lets the entire strut move around. Those are often next to impossible to identify until the strut is disassembled. That's when we see the defect and have to tell you more parts are needed than we originally knew about. New upper mounts come as part of those quick struts I just mentioned.
When the strut, ball joint, or control arm bushing allow the spindle to shift position, it lets the wheel and tire tip in or out on top more or less which is what changes the alignment. As it shifts, the steering linkage is connected to that spindle, so it gets tugged on, which is why the steering wheel changes position. Depending on the design of the specific car model, (some have the steering system in front of the wheels, but most have it behind the wheels), the steering wheel could turn in the same direction as the car veers during braking, but most veer in the opposite direction, as you described.
Here's another dandy article on this topic:
https://www.2carpros.com/articles/car-pulls-right-or-left-when-driving-or-braking
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Monday, December 28th, 2020 AT 8:11 PM