Welcome. I'm not sure exactly what your balance issues are stemming from, but one basic thing that might help is to look where you want to go. If you're looking right at the ground, and not far enough ahead, that might be a problem. Also, if there's something you need to avoid, like a pot hole, rock or other debris in the road, don't focus on it. Look for the way around it.
The other thing you might want to do is add some core work. A strong center will help your balance.
'02 Eddy Merckx Fuga, Selle An Atomica
'85 Eddy Merckx Professional, Selle An Atomica
'10 Soma Double Cross DC, Selle An Atomica
Slacker on wheels.