I'd say the bars are too wide, but I'm also not sure that alone is going to solve all of your issues.

My shoulders measure 38 on the nose, and yes, I ride 38" bars (Specialized Ruby's to be exact). I could probably be comfortable/ok on a 40" bar as well, but nothing wider. That's a totally different story on the mountain bike, but that's another post.

Going to something narrower (maybe consider 40's in your quest as well, but no wider, IMO) will improve the bike fit and feel.

What I'm not sure it will help is the brake lever reach in the drops. This sounds like more of a drop configuration and/or hand size issue to me (without seeing you). I solved this by actually going to a bigger drop (as opposed to the popular shallow drops) that allows me to really shove my hands all the way forward in it. Others, with hands of similar tiny-ness, seem to be perfectly happy with the shallow/compact bars.

In either case I'd recommend looking for bars that are short reach and have a short reach in the drops as well (that doesn't mean shallow top to bottom, but rather shallow front to back).

Also double check the brake lever positioning. Sometimes a slight tweak of the brifters downwards or a slight rotation of the handlebars makes all the difference in being able to reach them.

Good luck! and welcome!