Hi and thanks for using 2CarPros.
Honestly, I have no idea how engine temperature can effect handling. However, if the vehicle is hard to keep on the road, the first thing you need to do is check steering components. Also, make sure the tire pressure is correct. The colder the air, the more the air contracts. As it warms up, the pressure increases. So, if it is low, it could feel different between the time you start and it warms up.
As far as the steering components, safe lift the front of the vehicle off the ground. Firmly grab the tire at 3 and 9 o-clock and shake the tire left and right. You should not feel any play. If there is, have a helper watch for worn tie rods. Also, place your hands at 6 and 12 o-clock. Do the same to check for excessive play in ball joints.
Here are links that show how to replace a tie rod. I am adding it so you can see the component, and you will have the directions if you find one is bad. There should be no play in the tie rod. Pay specific attention to the part that attaches to the tire side (steering knuckle). That is where play will most likely be noticed. There is an inner tie rod that you cannot see because it is under a boot. If play appears to be coming from there, have the helper hold his hand on the boot to see if he can feel play.
https://www.2carpros.com/articles/tie-rod-end-replacement
Next, is the ball joint. Here is a link related to replacing one. The same principals apply to this as aforementioned.
https://www.2carpros.com/articles/how-to-release-a-ball-joint
Other than these things, you could have a loose rack and pinion, worn bushings, and many other things. However, the things I mentioned are most common and if bad require immediate attention.
Let me know if you have questions or need any help.
Take care,
Joe
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Friday, November 23rd, 2018 AT 8:50 PM