I measure 38 (according to my DH anyway
) but I need a minimum 40 cm bars, and I can ride 42s very comfortably. But if you're really 37 cm at the shoulders, 44 is likely too wide for you.
When you're in the drops, are your wrists much wider than your elbows? When you're on the hoods, relax your wrists - do they break to the inside? That would be how I'd decide whether bars are too wide.
But, I have small hands and absolutely need the short-reach brifters, positioned where I can reach them. As Martian said, reaching the brifters doesn't have much to do with the width of the bars.
Speed comes from what you put behind you. - Judi Ketteler