How can I stop high water pressure from causing leaks at the connection to my washer?
The usual remedy is to install a water-pressure regulator (aka. water-pressure–reducing valve) where the water line enters the building:



They cost about $70, and the plumber should be able to install it in less than an hour. But since you live in an apartment, this is probably something the landlord or condo management company would have to arrange.

There are also screw-in pressure regulators, commonly used with recreational vehicles:



Although I don't have any experience with them, it would probably work to use one with your washing machine. They are only ~$10, and installing it is something you can do yourself, even in an apartment.