Yes. 1962 was the first year for a convertible in the F-85 line up. There were 3,660 F-85 and 9,898 Cutlass convertibles produced in 1962. As you thought, from your description, your car is an F-85. If you look at the firewall/cowl I.D. tag an F-85 is model 62-3067 and the Cutlass is 62-3167. All F-85 models were 62-30XX and Cutlass was 62-31XX.
