Getting used to narrower handlebars Just switched from flat to drop bars which are narrower that the original flats.

Also, to compensate the "depth" of the riding position, I took a shorter stem (60mm instead of 105mm).

Because of these changes:


The riding hands position is different (which was the purpose, but I am not used to it yet): the hands are "vertical" when riding on the "safe" positions (on hoods and on drops) and are when in the "unsafe" position (around stem), they are very close to the stem which is unusual for me as well
The bars are narrower, so in any position, the hands are close to the center of the bars which feel very different
The shorter stem also affect the feel of the direction


As a result, in my tests, I did not have any problem per se, but my starts have showed to be a little "wobbly" (not riding completely straight on the 1 meter or 2) and on a general level I don't feel as safe as I used to.

I am sure a big part of that needs just getting used to, but still, any tips?
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.