Your 57cm ETT is about 12mm longer than the max. length French Fit, suggested by CC.

I'd say your frame is a cm or so too long, but you can work with that.

I would start with swapping the stem; perhaps look at starting with a 100mm stem.
You should be able to pickup a cheap one, to trial for a bit.

Seems to me, your pain is from reaching too far to the hoods.
Some may be coming from tilting your head up too much, when in the drops.

Note that drop bars come in different reach and drop measurements, also.

Yes, 2-3cm IMO makes a huge difference over the distance.

Lastly, from experience, going to a road bike position, your body needs time to adjust.
If you can afford it, I would definately pursue a good bike fit. Ask around in your area who people recommend.