How should I handle my insurance deductible cost after a hit and run when I took down the license plate number and I was not at fault Do I have to wait for insurance to go after costs or can I sue?

When in doubt...sue. I'm sure you have made a police report, press charges and then take the individual to small claims court for damages to your vehicle and any medical expenses, then notify your insurance company that you are suing the individual, they may pay the costs and pick up your lawsuit, try to speak with insurance representatives in higher-up positions if at all possible.