I'd of shopped and switched also but given the described events, any company is going to classify that as an "at fault".
You can petition the company that maintains the Industry wide database for a copy of your report and attempt to have the information removed, with "attempt" being emphasized. In the least you can provide your statement that will be attached to the report and sometimes can make a difference. That company is Lexis Nexis.
Years ago they (LexisNexis) listed an accident we had as at fault. We were rear ended while sitting at a red stoplight. We were able to get it revised only after many requests, eventually ending up in our attorneys lap to the tune of around a $500 bill and about a year. Our insurance at the time Allstate, despite not paying out as the claims were paid by the insurance of the guy who nailed us, were of no help but gladly raised our rates come renewal. It's a huge racket, and seldom sides with the customer.