It would be good to know if the current on that circuit's grounded conductor was equal.
Equal to what? Are you suggesting there may be a ground fault or a shared neutral? How would that show up as unexpected current?
I'm suggesting that knowing whether the mysterious 15a that appears on the hot wire is also seen on the corresponding white wire will tell you either that the current is a real load (or line-to-line arcing) or that it's a fault to ground or a grounded surface outside of the circuit conductors.
Troubleshooting is a process of elimination. Often, you have to find an intermittent issue by eliminating the possibilities one by one. You start with the most obvious or easiest-to-check suspects first, and whittle them away until you're left with that which may require replacement as a check.
In this case, when you find 15a on what should be an unloaded wire, knowing the complete current-flow pathway can help you determine where the fault is. Is it hot-to-neutral, hot-to-EGC, hot-to-metal-box-to-steel-stud-to-concrete-to-earth, etc? Is it in the wiring or a power cord?
Like the elusive noise the car owner can't get the mechanic to hear, if it happens while you're there, check everything you can think of immediately while you can. I suggested pulling the receptacle in case someone added another receptacle to the laundry, say outside or under the house.
A troubleshooting suggestion: temporarily wire a rubber socket with a high-wattage bulb in series with the breaker and the circuit black wire. The bulb will flicker visibly if you happen upon the fault while wiggling and moving various items. Unplugging the washer could break a fault, making it appear that the appliance is the cause.