first, a repair man that charged you before the problem is remedied is not what I would call reputable and I would question his qualifications as well.
in my 32 years of service work I have never charged for a part that the customer did not need, and never charge if I can't fix it. I would find another servicer.

as for your dishwasher not being grounded. the neutral wire, white wire, is always hooked to the same bar as the ground wire is. so unit is grounded as far as power is concerned.

check your wiring hookup at the dishwasher, make sure the white wire is attached to the white, and the black to the black.

if they are hooked up differently the polarity could be reversed and causing a problem.

next it is possible that you are having a power fall off. meaning the voltage seems to be there, but when put under a load the power drops off. you would need a volt meter to check this.

addressing the ground wire. it only needs to be there in case there is a short in the d/w. the ground wire would ground the frame of the d/w so there would not be a shock hazard on the metal parts you might touch. when you have a receptacle that only has two ports that receptacle does not have a ground wire running to it. things still operate because it uses the neutral wire as the ground.

I came into this late so I would like to know what is wrong with the d/w in detail. I will try to help