Yes, an ENT specialist is the correct choice of a specialist for you to see for vertigo type problems.
Vertigo with nausea is quite a common condition – your story sounds typical of either viral labyrinthitis or viral vestibular neuritis. These are caused by viral infections of the inner ear, or a viral infection/inflammation of the vestibular nerve respectively. Both conditions can be hard to tell apart as symptoms are similar, however labyrinthitis is more likely if you hearing has been affected too (vestibular neuritis does not cause hearing loss).
Less commonly, the above conditions can result from a bacteria infection too, which will require antibiotics treatment.
Another 2 possible causes are BPPV and Ménière’s disease – BPPV in particular is common above the age of 50.
None of these are a diagnosis for you however, as there are other conditions too that can mimic the above symptoms and conditions, hence it’s best to get an ENT specialist to have you checked out – they are the experts when it comes to vertigo/inner ear-related problems.
I cannot comment whether you need to see a neurologist at this point without examining you myself – but from what you’ve described alone, the ENT specialist will certainly be your first port of call. If necessary, they will also be the right person to decide if you need to see a neurologist.
I’m not sure if you are already being followed-up by an ENT specialist for your hearing loss too; but it’s unlikely to be related to your current episode of vertigo due to the time span difference.