Tinnitus and ear-related vertigo are caused by damage to different structures within the inner ear...but a truamatic event which damaged the sound sensing structures (cilia) could easily have damaged the semi-circular canals which provide attitude info to the brain. The celia and semi-circular canals are adjacent to each other.
It's true that a single gunshot should not cause any real damage...but that is for a gun held at arms length...if a 12 guage went off with the barrel near your ear, damage would seem likely to me.
If you're lucky, you may have incurred some damage to the eardrum, or the bone structures which connect the ear-drum to the tympanic membrane. Anything inside the tympanic membrane is harder to fix.
Sounds like you need to see a specialist asap. You should also contact an aviation medicine consultant to figure out how to deal with the FAA.