Old washing machine trips 20 amp GFCI outlet but never trips 20 amp breaker: ok to install 30 amp GFCI outlet?
Your choices: fix your washer's ground fault or get a new washer (oh, and get a new appliance technician while you're at it)

Your appliance technician clearly hasn't a clue, because the fact your GFCI is tripping and your breaker isn't means your washer has a ground fault, not an overcurrent.  So, you can either chase down and fix the ground fault in your current washer, perhaps with the help of an appliance technician who has a clue about GFCIs and how they work, or get a new washer.  (My money for a washer this old would be on the insulation on the drum motor windings breaking down after all these years, which means the motor either needs replacing or motor-shop work, but it could be some other sort of problem, such as a slow leak that's been dripping water into electrical parts somewhere.)