Yes you can request another adjuster. I JUST did with TD. Back in early Jul I was clipped on my highway bar by a road rager in my lane passing on the right, (Obvious dick he had both side mirrors tucked in so obviously drives VERY close to other vehicles). He then told his insurer I had hit him, (while lane splitting), and then fled the accident scene, (same reported to police). Unfortunately, for him I have a GoPro mounted front AND back of my bike and caught everythign except the side hit on camera, (but you can see the camera jolt, as he hit me, yes I stayed up)
TD adjuster calls and starts accusing me of this and that, and not reporting the collision. I advised it was reported to collision reporting center, and I called TD after and was told due to no damage no need to file a claim with TD. Long story short she was a b itch at best. I asked for a new adjuster. Her supervisior never actioned my request so I called customer care, (staffed by managers). In the end I got a supervisor to handle my "claim" (note the original adjuster had already assigned me 50% fault and it was attached to my insurance profile as well as listed as at fault on my DL abstract). I sent the supervisor a very short clip from both videos. She reviewed called me back 15 minutes later to advise TD had advised his insurer they were classing it as 0% at fault for me 100% at fault for him, and correcting abstract and insurance profile.
Now the real fun will begin, as I have obtained his info from insurer, and will be this week filing a claim in small claims court for damages, resulting from his false accusations and damaging statements.
So request a new adjuster.