Dog Bite Tennessee — How Do I Personally Pursue The Homeowners If Their Insurance Won’t Respond?

I’m looking for advice from anyone familiar with Tennessee personal injury law or who has dealt with a similar situation.
I was at a private property for a gathering with several other people. The family’s dog had been around me for most of the evening, and I had been petting him without any issues. Toward the end of the night, I leaned down to kiss the dog and he suddenly bit me in the face. The dog is a Cane corso has bit two others they didn’t report and the owners know he’s dangerous stating he has been aggressive since getting him
The bite was near my eye and went through my eyelid. The wound itself wasn’t huge, but I needed stitches and had to be treated by an oculoplastics team. I also developed cellulitis afterward and required additional treatment. My medical bills ended up being around $10,000.
I have an attorney, and we initially tried going through the homeowners’ insurance. The claim was apparently denied, but I wasn’t given a clear explanation as to why. There is another insurance policy that may potentially apply, but my attorney has been having difficulty getting a response from the homeowners/their insurance company.
The homeowners have since moved out of state, and my attorney has told me that I may need to pursue the individuals personally if the insurance situation doesn’t work out. I have addresses for the people involved, but I’m not sure what that process actually looks like in Tennessee.
My questions are:
If someone’s homeowners insurance denies a claim or won’t respond, can the injured person sue the homeowners personally?
How does that process work in Tennessee?
Is there anything I should be doing now to make sure I don’t miss a deadline?
I’m mainly trying to understand the process and what my options are. Also my current attorney has stated he doesn’t go after people personally.
Location: Tennessee but I live in different county than where it occurred (also in Tennessee)