Getting used to narrower handlebars
A big part of your problem is purely what you are used to and comfortable with, however wide, flat handle bars are used on comfort bikes and mountain bikes as they tend to offer more stability (at least, more perception of stability) than drops. I never feel as stable on my roadie on drops as I do on my MTB with relatively wide flats - although on the roadie I never screamed down bumpy technical single track at insane speed, so do not consider the relative instability of drops. However, hard to know if this is purely down to the bars, or the different the riding position, bike geometry etc.