I'd be interested to know how are your standing, walking and general use of your legs as a biped. Very often people think they have a 'balance problem' when they don't. The control system software and hardware that we use to move around on two legs is the very same as that which we use to valance on two wheels. Often all that's missing is a bit of 'translation' and this might be where the barrier is for you. If you can imagine yourself running in a figure of eight around two bollards you definitely have the skills required to ride on two wheels.
Trikes, recumbents, and recumbent trikes are great if that's what you need, but they are never as easy to live with as a regular bike. They're wider, longer, heavier so more difficult to store and transport. They're inevitably more expensive for a given spec than a regular bike and the options are very limited. You might struggle to find a bike shop which is prepared to service it, let alone supply one for you. So it's worth making absolutely sure that a bicycle is ruled out before turning your attention to the alternatives.
Your local authority recommended cycle trainer should be able to advise, they'll know how to get you rolling on two wheels (Bikeability level 1) if it's remotely a possibility for you.
nasty things.