Am I legally and financially responsible

What is the name of your state? Florida
Last year I was cut off by another driver and hit a crash attenuator. The other driver left the scene and I was left with a totaled car. The company that repaired the attenuator claimed that I was negligent and looked to my insurance company for payment of 14,138. Without trying to settle or getting a release for me my insurance company paid my $10,000 liability limit leaving me with a balance of $4,138. I was not at fault and did not receive a ticket, should I be held financially responsible for the damages?

You hit it, you owe the money. Unfortunatly, you don't the the other driver license #, do you? Any witnesses? Regardless of fault, your company has paid out on your policy for the repair. You didn't have high enough limits on the liability. You owe the money. Next time don't cheap out when purchasing your auto insurance. Higher coverage is only a few bucks more a month but can save alot a grief if this type of thing happens.