Bill,
Welcome to the forum. When you change front tires, the compound does have an effect but the size and design of the tire changes the dynamics of the steering at least as much. Some head shake is a common but not very desirable characteristic of trike conversion. Adding a 4.5 degree raked triple tree set like Champion's EZ-Steer will cure your problem and make you wish you'd done it when you converted your bike. Just ask anyone who has done it and you`ll get a 100% positive response. The cupped tire may or may not continue to negatively affect the way your trike handles after the rake.
BTW you may first want to have a competent bike shop re-torque your steering head and see if that helps if you aren't sold on the rake kit ...AND make sure your rear tire pressures are set at you kit manufacturers suggested air pressure which is normally around the mid 20LB range, not the 35LBS recommended on the automotive tire itself. These two can have a major affect on the handling characteristics of your trike.