Planning pacing in Southern Italy

There is nothing inherently stressful about southern Italy--there is something stressful about constant moving, and I would find moving every few days unpleasant. If you have a month, why not just start sketching out itineraries, and when you have a pace you like, cut what does not fit?
You could spend a whole month in Rome and not be bored.Sicily could certainly fill a month! If you want something more like a week in Rome, then I would suggest about a week and a half for Naples and its environs (at a minimum), then you could fill in with some time in Puglia. Then the next time, explore Sicily. Or, stay for three months :)

What we did, back in 2012, was stay Sorrento several nights, and see Naples and the Amalfi coast from there. We were coming from Sicily, so saw Paestum on the way to Sorrento. Being based in Sorrento for a time might cut down on the stress, with less unpacking/repacking.

Agree with Sorrento for a week...from there boat rides to Naples,
Amalfi Coast-Positino-Amalfi etc (can also access with very cheap
scenic bus rides along the coast), Capri and a charming town as a
home base...

Yeah, even for a Month, you are trying to include too much. I struggled with about the same issue, (also now retired) wanted to do Sicily, but some time in Rome again, and in the Naples area again, but also Puglia. We eventually dropped Puglia, saving that for another trip, doing a couple nights in Rome to adjust, then to Sicily, where we did Palermo 3 nights, Cefalu 2 nights, a night on the road, then 2 nights in Piazza Armerina, 4 nights near Agrigento, and 4 nights in Siracusa. We finished with 4 nights in Sorrento, and 2 more in Rome before flying home. As mentioned, we had been in Rome before (4 times, some 16-18 nights) and Sorrento/Naples as well, so we were not seeing every sigt in those places.

We enjoyed the 4 night stays, could actually relax a bit, could justify getting an apartment in a couple places so we could shop markets and fix a simple meal. I envision future trips will see longer stays in places, though we are limited to a month or a little more right now due to pets, property, and family obligations.

I guess my advice is choose between Sicily and Puglia, and if you have not been to the Naples/Amalfi area or even Rome, then choose just one of those. Target 3 and 4 night stays, maybe an occasional 2 night stay if passing through a minor place.

I should say that we're not really urban travelers. There are obligatory tourist sites in large cities, but we find them crowded and stressful. We have enjoyed smaller cites and town throughout Europe. They are much more relaxed and are easier to navigate. We can take all the time we want, but being away from home can be tiring too.

I like cities just fine, but I also prefer small towns and villages, especially when traveling in Italy. So, a whole week in Rome, Palermo, or Naples is likely not in the cards for you, but grab a guide book to determine how long you need in each to see your places of interest first. Then there are thousands of small towns in close proximity to each other, particularly in Puglia, so you can prioritize longer stays in those. The cities will form the backbone since you'll fly into them or arrive by train, then you can work out the small-town part, ideally with a car for Puglia and Sicily (Not using the car for cities but in between). The Amalfi Coast is a snap to explore without a car, as there is ferry and bus service linking the villages.
I agree, either stick with the mainland, or combine Rome/Naples(and area)/Sicily. Transport-wise, that works well. Since you like smaller places, you might want to look into Puglia's neighbor Basilicata too.
Lucky you--this is certainly what I look most forward to about retirement!

Am I correct in assuming this will not be a summer trip?

We like to travel in the shoulder season or off season. It avoids crowds and is less expensive.

We did Puglia and Sicily together flying from Bari to Palermo on Volotea. You need more than two weeks for Sicily.
Two to three days in a city would be a blur. Your instincts are right. Either
Rome, Naples, Amalfi and Pasteum or Puglia and Sicily would be a huge
improvement in your schedule.

Spend the month in Sicily. We were there in June 2019 and spent 3 weeks. We did not see all of the island. We saw a lot and had a nice pace. If I am reading your post correctly, you want to travel to all the places you listed in 1 month.
We did another 3 week trip of Naples, Amalfi. Matera and Puglia. With that said,
You could do 5 nights Rome, 3 nights Naples, 5 nights Amalfi, 2 nights Paestum, 7 nights Puglia ( which will give you a couple of bases ) then 7 nights in Sicily. That will give you 2 areas in Sicily maybe fly into Catania spend 3 nights Taromina and 3 nights Ortigia and last night in Catania for flight home.
The one thing you did not mention was when you are planning on going.

There are obligatory tourist sites in large cities, but we find them crowded and stressful.

Cut out Rome and Naples then decide between the Amalfi Coast / Puglia or Sicily.

That's a lot in a month. Personally, I would save Sicily for another trip.

When you say Puglia, are you talking about RS Puglia which is basically just Alberobello, or really seeing Puglia--Lecce, Otranro, Gallipoli, etc.?

You can easily make this a circle trip without having to backtrack.

A month is tight to cover all the way from Rome to Sicily. I would do either Naples to Sicily, or Rome to Puglia, with a strong preference for the second option.

In the second option, I would allocate a month as follows:
- 5-6 nights in Rome
- 5-6 nights in Naples incl. Pompeii and Herculaneum
- 4-5 nights around Amalfi coast including Capri
- 3 nights for Paestum + Cilento region (best with a car, rented in Salerno)
- 1-2 nights in Matera en route to Puglia (keep the car)
- 7-8 nights in Puglia (keep the car), either in a single base such as Lecce (accepting you will not be able to visit the whole region), or in two bases e.g. Lecce and Monopoli or Polignano a Mare
- Final night in Rome before flight home (take the train from Puglia). Or if no good flights from Rome to home, fly from Bari to anywhere convenient.

I suggest a car for the 2nd half, but it is very much optional.

We've spent about two months total in Puglia (in March, April, and November) and it is the opposite of stressful. We rented apartments in towns or stayed in trulli in the country (but near towns like Cisternino and Locorotondo) and easily drove everywhere. You could spend 5 nights to a week in different locations. Some suggestions are Trani, Ostuni, Lecce, and somewhere near Bari like Giovinazzo. Include Matera in Basilicata (3 nights is OK here). Visit Alberobello for a couple hours.

You could spend a week in Rome or Naples before or after Puglia (which we've done) or probably the Amalfi Coast (but we've never been there). Sicily needs its own month-long trip.

Rick Steves needs to spend some time in Puglia so he will exclude it when he talks about "chaotic" southern locations. People go there to relax and slow down. The interior of Campania is not stressful or chaotic, either. Check out, for instance, Sant'Agata de Goti.

Our thanks to all who commented or made suggestions. It seems that we must scale back on our travel ambitions and enjoy longer stays. Less can be more. This could be two trips, or three, or a much longer trip. Now we need to decide what is most important to us.