I had similar problems when I got my first road bike in 30+ years back in 2017 (a Centurion Ironman, standard 1980s road bike like yours). I had chronic pain and poor flexibility in my neck and back due to car wreck injuries several years ago.
At first I made lots of little changes to make the bike a little more comfortable. And I did lots of physical therapy at home and in clinics -- it's still a daily thing to improve or at least maintain my flexibility and strength. Between the bike and body changes I'm now able to use the drops for a minute or so at a time. On the hoods and tops, no problems on rides of 20-50 miles.
And I needed a better saddle. The 1980s style Turbo saddle and copies don't suit me. I need a flatter saddle, fairly narrow (130mm), longer nose and prefer a combination of moderate padding and flexible plastic shell. The newer Selle Italia and similar saddles work best for me. Being comfortable in the saddle affects my comfort on the handlebars as well, and vice versa.
Here's a fairly recent photo of the bike -- I think only the saddle has changed since I took this photo.
The major change was from the original 120mm stem to a 90mm. Much better. It did change the handling a bit so any stem change will require some adaptation. Usually shorter stems make the bike feel a bit twitchy at first. If you can see the front hub while in the drops or on the hoods, the stem is probably too short. If it's too long it will feel more stable but more stretched out which can strain the neck, shoulders, etc.
I also tipped back the original handlebars just a bit, so the bottom of the drops is no longer parallel with the ground. Nothing extreme and it puts my wrist in a more natural alignment.
I raised the brake levers/hoods back just a bit too. I'm on the hoods most of the time so this suits me. I adjusted the brake lever reach (via a grub screw under the points of the brake hoods) so the brakes are fully accessible from the hoods or drops.
The blue tape on the black stem is a makeshift witness mark indicating my previously lower stem height. After I was hit by a car (again) in May 2018, re-injuring my neck and shoulder, I raised the stem about 1/4" and found it more comfortable. I planned to lower the stem as my fitness improved but it works so well in that position I've left it as-is most of this year.
If you look at photos of professional grand tour racers in the 1950s-'60s they often tipped the brakes back so far they were flared out too far to reach the levers from the drops, with brakes that did not have levers with adjustable reach. But in that era the emphasis was as much on comfort as speed and power over the course of a three-week race, especially if they planned to participate in more than one GC race that season.
1989 Centurion Ironman. Mostly original with several minor adjustments to suit my comfort and (hopefully improving) conditioning after several injuries to my neck, back and shoulders.