Centering moving road with no reference point

The last thing you want to do with that shot is run Warp stabilizer. It will not gain you anything and it will probably distort the lines on the road.

What you want to do is Motion Stabilize the shot so there is no movement in the frame. Before you Motion Stabilize you want to trim the shot so that you are only working on the frames that will be in the final edit.

I don't have time to generate some screenshots to show you how I would handle the project right now, but stopping the camera motion, pre-composing, then positioning and animating the pre-comp is the way to go.

To stabilization, try the warp stabilizer. (Thank you, Rick!)

Or you can track one of the white lines and use tracking data for stabilisation. You only need to lock x-axis and you have to readjust the tracking point several times, because the white lines are moving out of frame.

If you mean by "vertically straight" to disort the street, so it looks straight, try one of the disort effects, but don't expect too much of if. The road has a bend and only detailed image retouching will make it believable straight.

*Martin