DubyaStep

I suggest installing a new handling line to better fit the model you are using as the vanilla handling lines are not fit for some modded cars. '?do=embed' frameborder='0' data-embedContent>>. There you go. Hope this helps .

I would go with the advice above given by Jewish Banana. (epic name)
 
But, there is also another way....  But take note, unless you know what you are doing you can seriously screw up your cars handling so much that it flips over by by breathing to close to it.
 
Any google search for GTAIV handling editor will throw up a couple of different but both very usefull tools that enable you to tweak everything you can think of.  From steering lock and suspension stiffness. To top speed, gears and torque.
 
There is a 3 part video tutorial that explains (mostly) what each of the settings do. The vids center on how to make cars drift. Ignore that. It's just the explainations of what the settings do that's important.
 
First part of 3  http://www.youtube.com/watch?v=5askSm7kjXU
 
But once again be warned. This is advanced.....
Also a real life understanding of car physics goes a long way to understanding some of what these handling editors do.
 
Edit: Also take note that those 3 vids contain a huge amount of unwanted waffle. It could all have been explained in one vid. But there ya go.

Unless any othe cars you installed came with a handling.dat file, or files that made changes to that file then installing a certain car such as a taurus wont affect another car such as your cvpi.

I think somewhere along the line you have changed a handling line without realiseing it.
I can't (at the moment anyway) think of another explaination.

One Option for the runway thing, enable never wanted in the trainer menu.  

Try logging out of Games For Windows Live and then deleting your savegames.

That's the problem. Unfortunately you'll have to delete your save-game and start a new game; that should solve it.