I believe we have similar dizziness. Not room spinning, but more imbalance feel like falling, head presure, external stimuli invoking dizziness mostly sound etc.,
I found by chance after few months Xanax (another benzo) helped me a lot. If only they prescribed a benzo or some anti-seizure med instead of Elavil it would have helped a lot. Then some ENT doc switched me to Klonopin but it affected my sleep and I woke up with sleep apnea every morning which made my life very difficult.
But Xanax helped me a lot and now I taken whenever I am more dizzy than usual and have to drive some distance. It also relieves my head pressure and drug gets in and out of your system quickly. I take it during the morning or afternoon and never in the night. Drug induced sleep doesn't help me. Benzos are commonly prescribed for dizziness if nothing else works. I know some people with true Vertigo take it when they have a bad spell.
Doctors never mention Benzos unless you are in a terrible condition since it is highly addictive and some get used it very quickly and need more and more. It works like alcohol on GABA and alcohol too helps me but I don't drink. Xanax is the number one prescribed drug currently in U.S
How well it works for dizziness is a trial and error. If you think you can take on a need only basis and feel terrible then you can try. If not stay away. I have some dizzy feeling and pain 24/7 so I do get tempted to take a bit of Xanax but I learnt to control. I know it is a tough call but only you know your suffering. Have you tried Dramamine? Not much medicine out there for dizziness.