Either states will decide you don't need insurance if you have a self driving car, or a company will spring up that will insure self driving cars for a lot less money.
It is one area where capitalism can work. Lets say all the existing insurance underwriters charge $100/month for normal insurance based on human drivers. At that rate they can cover the rate of claims and make a nice profit. Say $20/month ends up being net profit after their operations costs and payout are factored in, and operations are another $20/month.
Well lets say that self driving cars then have a 0.01% accident rate compared to human drivers (it may end up being lower than that). That will drop their payouts by a similar amount, so from $60/person/month to $0.60/person/month. Ok but they decide to keep the price the same, just make more money.
Thing is, they'd still be really profitable at $41/month, instead of $100. Someone else will realize that, and work to steal their business. They might not go that low, maybe $80/month, but it'll happen. Then they'll try to get it back and so on and so forth.
Remember that your costs aren't just based on your specifically, they are based on actuary data of accident likeness. Sure you've had no accidents, but there is a statistical probability that you will. You are in the lowest risk group likely, but it is there. If self driving cars are much lower, rates can again be much lower.
Also, have you checked around? My rates haven't gone up in a long time. Maybe your company is just screwing you because they can, and you'd save if you took your business elsewhere.
For comparison purposes I pay about $350/6 months for $200k/$500k liability insurance on an old, cheap, car.