Unless you took the skills test for "A", you're not qualified to drive that class. I would go back to the DMV and have them verify everything. Driving a vehicle you're not qualifed to drive is an "OOS" offense and a fat fine.
When you took your class B skills test (backing and road driving) did you have a trailer or just the straight truck? If you had a trailer you may have qualified for a class A. If you did not have a trailer you got the wrong license.
Firefighters operating commercial equipmentowned or operated by or for a government agency are exempt from holding a cdl. If these individuals chose to operate one outside of the agency,they are subject to the same rules as you and I.