Riding a road bike feels like trying to balance on a ball

[vkai]

I've ridden a Trek Navigator for the past few years. My Navigator is too big for me, but I've managed. Off and on I've tried road bikes, but I've never been able to feel steady on them. I'm not sure if it's the new position or what. The first thing that came to mind was that the bikes I was trying were too small (I feel very shaking riding the bikes), but I can't stand over anything taller. I'm 5'4" and I have short legs even for that. I've never really gotten the hang of mounting and dismounting, so I know that doesn't help. Oh, and I'm scared of my LBS.

I wonder if it's because of the tiny little tires and my 240lb body being supported by them.

I wonder if part of it is due to generally weak arms, but I broke the left one four years ago and had two surgeries two years later to repair and subsequently remove some cartilage in my wrist.

I wonder if I am just trying the wrong bikes.

I wonder if part of it is from being used to riding a bike that's too big.

I feel like I'm putting too much pressure on both my hands AND my butt. That can't be! (okay, maybe that's just my weight)

Anyway, I was wondering if any of you have any advice. Is that unsteadiness likely due ot something specific? Is it just a matter of getting used to the bike/position?