You will need to contact the Department of Motor Vehicles for the state of Washington for information on registration. If you are required to register your car in Washington, then you would likely need a Washington insurance policy with your Seattle address.
Since your son lives in his own home at college, you will not have to add them to your policy if they drive your car less than ten times per year, with most companies. Some insurance companies offer a discount if you add a college student who lives away from home and has limited access to the insured car and seldom drives it. Every company is different so I would check with your company for their specific rules.
You will need to transfer the title of the vehicle to the new buyer. It is important that you fill in all of the appropriate areas required in the title certificate when selling the vehicle to another party.
The problem would be with the state of Rhode Island concerning your registration. If your vehicle is no longer insured in the state then you will be expected to turn in your registration tags and get new tags in the New York.
Pay the ticket! Violations issued by police are reported to that particular state's DMV. An unpaid ticket can create all sorts of issues from a suspended license in your current state, inability to renew an expired license, and even a warrant in the state where the ticket was issued. There are instances where a ticket fails to be passed on to the DMV and never gets reported, but those situations are rare and the potential risks of not paying far outweigh the gamble.