Can too long a reach create problems? Sure. Do you feel really stretched out on the bike? Do you feel comfortable on the handlebar tops but have trouble reaching the hoods? When you are on the bike riding around and you put your hands on the brake hoods, is there any bend left in your arms, or are your elbows straight and your arms reaching?

Also, you may also want to notice whether your shoulders are creeping up towards your ears as your ride. You don't want to hunch up your shoulders.

One of the reasons I asked about the height of the handlebars is that you say you have more trouble in the drops than on the tops. Being in the drops may cause you to have to bend your neck back more. It could be the bending of the neck to look up, especially if you have weak deep neck flexors, that could cause your pain.

Also, if you wear glasses and have to lift your head even higher to see things through the glasses, that can be a problem.

Of course, there are also other things that can cause the problem. For example, your upper arms should make roughly a 90 degree angle to your torso, but this is hard to see without seeing you ride. One of your best bets would be to get some in-person assistance at a bike shop or from a bike fitter. They could give you a sense of whether your reach is too great.

A 59 cm bike is not typically too big for your 6'3" height. You may have to adjust the bike a bit, though. As far as reach, you could get a stem that is 110mm instead of 125mm, but someone knowledgeable who can see you on the bike would be better at deciding that.