The GFCI randomly trips when washing clothes

A GFCI is designed to trip only when it detects a connection between the ground wire and the hot line of the AC power line. So the fact that it's tripping on a new appliance would generally mean one of two things; either there's an electrical fault in your brand new washing machine, or your GFCI outlet is faulty. They do go bad; I've had to replace one in an friend's house that was acting up and tripping for nonsensical reasons, after which everything was good.

Did you have an old washer connected there that worked fine? Swapping it back in is the only thing I can think of to test an overly sensitive GFCI outlet, but barring that I'd be tempted to just replace the outlet and see if that fixes the problem. You could, of course, replace it with a standard non-GFCI outlet, but that probably wouldn't meet code and you'd lose the GFCI protection, but then again I've never had a GFCI outlet on my washing machine and there have been various machines plugged into the same outlet over the last 30 years without a problem.