Craig,
There are many areas to address before getting into tire shaving. That would be the last thing i would do after a thorough appraisal of your front end.
First, place a floor jack beneath the spring pans on each side and check for any play in the wheels. Any wear in the king pins, tie rod ends, or wheel bearings will result in shake at speed. Also check for wear in the splines of the hubs and wheels.
Second, check your alignment. Toe is the only adjustment available on a big Healey, and it's not difficult to do by yourself.
Third, check the balance of the wheels. You say you have new tires, but did the shop that installed them properly balance the wheels? The wire wheels on our cars must be properly mounted to the balancing equipment, and not every tire shop takes the extra time necessary.
Congratulations on getting your car on the road. It is a great driver when all is properly set up.