OP,
You have so many fit issues going on. The only thing I can tell you again, speaking for me, your fit is way off. This is no big revelation as fit really takes time and a work in progress. You are a perfect candidate for a pro fitting and I am not one to recommend a fitting unless you need to start at sqaure one.
Many things don't add up. For example your CC on line nos. don't agree with a guy who is 6'1" ish. It is easy to make a mistake with measuring your body and plugging in values. Have your wife or girlfriend do it 2-3 times and then take the mean of the nos. We are about the same size. Further, I would have a lot of pain riding your bike and I ride a lot without pain. You have a lot of saddle to bar drop which is uncomfortable for many recreational cyclists including me. The bike is really too small for you as the CC calculator stated...nominal you should be on a 59-60. Problem is for your leg length is...your head tube is pretty short making it hard to get the handlebars higher to take weight off your hands.
You own a racing bike geometry and you are just starting out. I would suggest a comfort geometry bike if you want to get more comfortable. So consider at least a change in type of bike if your goal is to become more comfortable and ride more miles.

A quick explanation of saddle setback and why its counter intuitive. When you get out of chair, notice that you will always stick your bum out. If you don't, you will have a hard time getting up. The same is true of riding a road bike. You feel stretched out because your position is all wrong. When you push the saddle more behind the cranks your weight shifts back away from the bars. When you fold your torso over to get back to the bars you need a longer reach because your torso is more horizontal. Basically with a more folded body, your prodominant body mass is further back. The notion of core strength is often debated here. Road biking doesn't require excellent core strength if your weight distribution is proper on the bike because your body is balanced on the saddle..which is the starting point for most fitters. Further your hip flexors restrict your weight falling more into the bars with better posture. A good posture on the hoods is with about a 45 deg torso angle from vertical. Better riders ride with even more torso angle on the hoods and close to a flat back in the drops.
Below I have included a picture of an ideal recreational fit...what I call an out and up position. Take a picture of yourself on the hoods leaning up against a wall. The rider has a nice stretched reach to the bars, in a relatively areo position with weight back and little weight on his hands and not much drop. This position works the best for me. I have also included an early pic of my Roubaix build with the bar and saddle height close to the same which is dramatically different than your bike. I ride in a position close to the rider shown with a good reach to the bars.
Basically you are riding 'exactly the opposite' of how I ride...your position with a handlebar that is close in and much lower. Further you are riding with less setback which places your weight forward and more weight on the hands. So you end up with worse case for weight on your hands.

Learning about your fit on the bike is a first step to becoming a good cyclist that can ride fast and in comfort for many miles on end. I will caution you however, as you learn and improve your fit, road biking will become more addictive.