That was super-nice of our member, who works in insurance, to offer help!
Glad you are OK! You did yourself a big favor by taking photos of the vehicle while it was still on-scene. If you managed to get photos of the other vehicle and the surroundings, that information will be really helpful to your insurance company.
I was in law-enforcement on a limited, part-time basis for a while. I do know that (in my community) citations can take some time. Tickets may not be issued by the officer on-scene. They may be issued after the traffic supervisor reviews the reports, which can take some time. Without the ticket, your fight is a little tougher, but not impossible.
Did anyone witness? Were statements provided or at least contact information for follow-up?
Witnesses can make the process so much easier to help determine who is at fault.
If no ticket and now witnesses, then the two insurance companies will review the reports, statements of both parties (and your photos). Since neither company wants to pay out, both sides will probably do their best to determine which side was responsible. Based on the location of initial impact to the vehicles and the position of the vehicles were when it happened, the investigators might be able to make a determination who was rightly entering the intersection or who failed to stop or failed to yield.
I found this on Progressive's site, if it helps explain some of what goes on...
The state where an accident occurs, and whether it's an at-fault or no-fault accident state, has a significant impact on how an auto insurance claim is paid.
www.progressive.com
MA is a no-fault state. I rear-ended an SUV after it left the ramp to enter a traffic rotary (translation: a huge over-sized roundabout for you non-New Englanders...). Nobody stops on a rotary! I was not cited. 18 months later, I finally got my appeal hearing. I was still found at fault. My bill has been high for 2 years now. 2 down, 5 to go before rates get back to normal (which are still STUPID HIGH).
Good luck!