Mark is probably correct. [on edit] Jerry also got there first (as usual.)
Loose steering/suspension components could cause that. With a different load on those components with brakes applied, the steering geometry could be affected. A Model A I had would sometimes exhibit "death wobble", but applying the brakes would eliminate it. This would be similar but opposite. Maybe toe-in is changing enough with brakes applied to cause the wandering.

Last edited by klhansen; 11/18/2020 9:10 PM.

Kevin
1951 Chevy 3100 work truck
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1929 Ford pickup restored from the ground up. | 1929 Ford Special Coupe (First car)
Busting rust since the mid-60's
If you're smart enough to take it apart, you darn well better be smart enough to put it back together.