Yes, you do. You had it 98% solved in your original note.
High-school geometry suffices. Extend each axle to be a line not just a
line segment. If necessary (to account for different-radius wheels)
consider the orthogonal projection of each line onto the plane of
motion. The point of intersection of these axle-line-shadows is the center
of the motion. What's the problem?
Tilting of otherwise-ideal wheels does not even cause any trouble.