I However how DO I get a bike to stay upright long enough? Every time I get on a bike it is just a heavy lump of metal that just wants to fall over, preferably taking me with it. I'd love to be able to practice "long swoopy curves"!
Riding a trike is totally different to a bike, on a trike to steer by steering and balance by sitting within the 3 wheels.
On a bike if you want to veer right you shift your weight right. It is a very subtle shift, mainly from your stomach and hips, ie your centre of gravity. As the bike leans in the weighted direction, it will fall over unless the wheels catch up and get underneath the centre of gravity again. This is where you steer into the curve, ie into the lean. Try pushing the bike along as you walk, leaning the bike and recovering balance by steering.
You have to let your body be very responsive, so relaxed and mobile. Dont hunch up and tighten your shoulders.
You probably need a learner bike, ideally a step-thru singlespeed that is smaller rather than too large. You may benefit from someone pushing the saddle as you balance.