Give it time…. I have recently purchased a croix de fer 30 for my 24 ish mile round trip commute. Like you I had never previously been on a road bike and never used drop bars before.
The first week was not great. My hands hurt badly especially at base of thumb. I was getting shaken about on rough stuff and was actually seriously questioning all that I had read about cx / touring bikes being good off the road.
Second week in, I fitted a zero rise stem 10 mm shorter and rotated the bars so the long bits of drops were parallel to ground. Such a small tweak made the difference between the bike fitting and feeling like turd.
Hand pain disappeared overnight.
Now it’s my go to bike… Happy doing loops of blue / red Swinley as well as usual commute.
Recently got some 40 mm marathon supremes which are great in Road at 70 psi. But drop to 50 psi and they are so comfortable off road.