Sure if you want flat bars, get them - lots of people put flat bars on road frames!

CyclChyk, I think everyone else was just trying to address whether flat bars will actually help you. You didn't sound sure of that. Your OP wasn't clear as to exactly what kind of trouble you're having.

Also, sometimes knowing what to do isn't the same as executing it. A few years ago I completely lost all confidence in tight turns on my motorcycles. I used to be able to do them just fine. Now, I just can't. It's completely a mental block. I need to practice lots more, but whenever I try to practice it seems counterproductive because I get into such a panic state, don't seem like I'm making any progress, sometimes wind up dropping my bike. Has nothing to do with the bikes and everything to do with ME.

So. I think we do need more information from CC as to what specifically is giving her trouble. Since she says she's perfectly fine on her MTB, I don't think it's visuomotor issues.

The first question I'd ask is how long have you been riding the Bianchi? You said you had given it some time, but you didn't say how much time or how many miles. I do remember when I got my race bike, and even switching from a touring road bike to the tighter geometry, it felt incredibly twitchy and unstable. Unfortunately it was over 20 years ago and I don't remember how long it took me to adapt.

Then I'd second the recommendations for a good fit. That's true whether you keep the drop bars or make it a flattie. How long is your stem, and how does it compare to your MTB's stem? If the stem's too short, it can make steering very, very twitchy. Unfortunately, IF you've installed a very short stem because your top tube's too long for you, the frame may just not fit.

Third, take a bike skills course where you practice handling and tight turns in a controlled environment.

Fourth, work on core strength. If the bike's a good fit but you still find yourself putting too much weight on the bars, that could be a reason the bike wouldn't feel balanced.

I don't think anyone's telling CyclChyk NOT to put flat bars on her bike. Lots of people do it. Right now I'm entertaining thoughts of turning my old race frame into a commuter, and if it happens, flat bars are a possibility. We're just not sure that flat bars will address the trouble she's having.