Are you supposed to sue the insurance company?

A couple months ago I got into a car accident with my boyfriend. It was his fault and I ended up having to go to the hospital. His insurance covered me up to $5,000. The bill is crazy though, already the cat scan was almost $3,000 and the bill is over $4,000 so far.

His insurance is pissing me off because they haven’t taken care of the bill after I called many times. Anyways, a couple months later I’m still having pain in my shoulder and back. I don’t have health insurance and I don’t think my bf’s insurance will be enough to cover medical expenses for treatment because it’s already almost $5,000.

I haven’t received any treatment. Many people told me that the insurance company has to take care of my medical expenses and that if they don’t I have to sue them. I’m worried about my boyfriend having to pay higher insurance though, I’m worried about it affecting him. Is it wrong if he pays more insurance due to covering my medical expenses? Should it be something I have to pay for my own??