At-fault accident, damages exceed my policy limits — what actually happens?

Hi everyone, I’m really stressed and hoping to hear from people who’ve been through this.

I was recently in a car accident and my insurance has accepted full fault. I have $15,000 in property damage coverage, but the other vehicle’s damages are about $27,000, so there’s roughly a $12k gap.

My insurance said they’ll pay up to my limit, but I’m worried about what happens with the remaining amount.

For anyone who’s been in a similar situation:

1. Did the other insurance accept the policy limits, or did they come after you personally?
2. How often does this actually turn into a lawsuit?
3. If they do pursue the remaining balance, what does that process look like?

I don’t have major assets, which I’ve heard can matter, but I’m still really anxious about the possibility of being sued or owing thousands out of pocket.

Any real experiences or advice would mean a lot. Thank you.