Can the tire type and profile significantly affect balance?
I did the exact same thing a long time ago. I wonder what width tyres you used?

Basically, I had a touring bike which I used for "rough" touring, and used the 38c version of this tyre. 

It was fine at lower pressures in rough roads. 

I then bought a set of 25c versions for a winter road bike. Horrible! These, tyres are not designed for smooth roads and high pressures. If someone doesn't understand this problem, basically, the tyre has no smooth corning edge. None. 

Just replace the tyre for something more friendly.

We can recommend tyres for specific scenarios, but you haven't mentioned the type of riding or your preferance in setup.