Is important to have the vehicle aligned when replacing the tires

This depends on what is needed to be aligned. You have three important angles caster, camber, and the toe. Caster being out of spec does not cause tire wear so this is the least important in relation to tire wear.

Camber is angle of the tire of the top and bottom tilting in or out away from the vehicle in relation to the center point of the wheel or the hub. This will cause inner edge of tire wear when it tilts too far in toward the vehicle. Outer edge tire wear when it is tilted to far out away from the vehicle.

Toe is when the leading edge of the tire is pointed further in or out in relation to the trailing edge. This will cause abnormal wear pattern across the tire.

Here is a guide and document that will help with this. Thanks

https://www.2carpros.com/articles/how-car-tires-work

Now, with any of these issues stated here, the driving will be off a bit. In the sense that the car can sway?

Yes. Any of these angles if off enough will cause the vehicle to pull to one side of the road or the other.

When you say the vehicle sways that normally means more of a wander. Meaning the vehicle wanders back and forth on the road. So it feels like it is pulling one direction and you correct it and then it is pulling in the opposite direction. This is normally an issue with the steering system. So I just wanted to clarify that.

I had a recall on the shifter of the vehicle. The Toyota dealership fixed the problem and also diagnosed the entire car. They found 2 things: the muffler pipe had an exhaust leak,which I replaced with a new one, and the inside of the shifter had a small crack which they said it should not be a problem as long as I don't shifted to hard.

Okay. Neither of these issues would contribute to a pulling or swaying issue. Let me know if I am not understanding what you are asking. Thanks

No, I know. The reason I informed you about the diagnosis on the car is to let you know that they only found 2 issues on the car. Should they have found this 'pulling" or 'swaying issue when they diagnosed the car.

Oh. I understand. I would not expect that they find that. However, "should" they find it, yes. They should perform a complete inspection of the vehicle which includes a test drive and if it is pulling then they should find it. However, I would not be surprised if they didn't. If you tell them that you feel it is doing it, then they will drive it looking for it but I have found most mechanics don't pay that much attention to it when driving a vehicle.

I see. Thanks. Would I need to worry about it?

If the vehicle is pulling and you have newer tires on it then I would get it aligned. If the tires are in need of replacement or close to it, I would wait until you put new tires on it.

Okay. Thank you.

You are welcome. Thanks for coming back to 2CarPros. Talk to you next time.