Road bike very twitchy when standing in the pedals?
Riding out of the saddle is a totally valid way to ride, you may just need to practice more. I remember having much the same reaction to my first road bike.

Almost all road bikes are twitchier/have faster steering than mountain bikes.

Mountain bikes are typically designed to be very stable to stop ruts, roots, and rocks from throwing you into a ditch or off of a cliff.

Road bikes are designed for nice flat, smooth tarmac and don't really have to worry about these obstacles.

Even within the road bike gamut, you'll find that the stability can vary wildly. A bike for touring will be very stable but one designed for criteriums will likely be the twitchiest of the lot.

I can't comment on the sizing, if you're test riding at a shop, why not try something bigger or smaller?