Very familiar problems. While I'm in better shape now than I was in 2015 when I resumed cycling after 30+ years off, I still have problems with neck pain from an old injury, aching wrists, etc.
Tire pressure is the first thing to try. The maximum tire pressure on the specs usually feels harsh. Costs nothing but a few rides to try. There's a sweet spot where slightly lower tire pressure yields a much more comfortable ride without sacrificing much to added rolling resistance.
For example, I weigh 160 lbs. For the Michelin Protek Cross Max 700x40 tires on my errand bike, Michelin recommends around 75 psi for my weight, up to 85 psi. That's way too harsh and jittery. I prefer 'em around 50 psi front, 55-60 psi rear.
Ditto the 700x42 Continental Speed Rides I used on my other hybrid for two years. The recommended pressure was around 75 psi. That's way too high for those tires. They're best around 40-50 psi front, 50-60 psi rear, depending on terrain -- lower for gravel and off-road, a little higher for pavement. They never feel sluggish at lower pressure. Highly recommended tires if they'll fit your frame (they actually measure closer to 700x38, narrower than the specs claim).
I recently switched to Continental Sport Contact II, mostly out of curiosity and to fit my old fenders. The 700x32 tires are much narrower than the Speed Rides. They felt way too harsh at 80-90 psi. Around 60 front, 70 psi rear feels about right -- comfortable without feeling sluggish.
After finding the right compromise in tire pressure, you might consider a few other cost effective equipment changes.
Grips are easy to swap and can make a huge difference. I just switched from a set of old foam cylindrical grips to a newer set of firmer but wider grips. I twist the palm support until it feels about right (using isopropyl alcohol under the grips to ease twisting, then let it dry overnight before riding). Spreads the road vibration over a larger surface, much more comfortable yet still feels secure -- not excessively soft and padded. Cost a whole $10 or so. Ergon grips with locking collars cost a bit more but you get top quality.
Small changes in stem/handlebar height and reach can make a big difference in comfort too. A small 1/4" change here and there may be all it takes. Easy with older style quill stems, a bit trickier with threadless stems -- your local bike shop will have spacers and can help you, or you can order 'em online.
You might also try another handlebar. Flat and riser handlebars are great for maneuvering, but not so comfortable over longer rides. Over the past couple of years I've switched my favorite hybrid from flat to riser and now to Nitto albatross bars with swept back grips -- a very traditional look but perfect for me on that bike. No more aching wrists, elbows, shoulders or neck after rides of 30 miles or longer. And I switched from thumb to bar-end shifters so the controls suit the new bar.
But depending on your bike and preferences, a different style bar with swept back grips might work better -- North Roads bars, etc. Some folks even flip their bars upside down for a more aero profile but it is a bit more strain. There's also the Velo Orange Belleville bar, swept back but with no rise -- a flat bar with lots of arc. Puts the wrists in a more ergonomically friendly position. And you can lean into the forward part of the bar for a more aero profile when riding into head winds, climbing hills, or just to change hand positions occasionally for comfort.