Lucas has hit the high points, and correctly cautioned about June hiking. The later the better, especially since there was a lot o f snow this year.
June 15th is an average date ( with wide variations) for this nice trail at about 2000 meters to open:
Altitude is the issue. Since you are experienced hikers, you probably use topographic maps. This one will give you altitudes. It has a good search function.
map.geo.admin.ch/?topic=ech&lang=en&bgLa yer=ch.swisstopo.pixel karte-farbe&layers=ch.bav.ha ltestellen-oev,ch.swisstopo.swiss tlm3d-wanderwege&layers_visi bility=false,true&E=26 37495.89&N=1160623.16& zoom=5
I have turn on the option in the menu to show hiking trails, which are color coded according the description this link:
I have very limited experience in the Dolomites, so I can't really compare. I can say that my experience with Italian trails,maps, and trail markings was nowhere close to the quality of trail marking and maintenance that you will find in Switzerland. Not sure about Austria, but I have some impressions that it is pretty good.
I can tell you that the Jungfrau Region, to name one of many, in Switzerland has a wide range of public transport, hikes for all skill levels, nice lakes, and stunning scenery:
www.swissholidayco.com /Public/Assets/User/fi les/Map-of-Jungfrauregion1.jpg
<<"towns we should be sure to go through?">>
Even though I've been going there for a while, I found this report to be extremely useful, as have many other forum members:
It will answer a lot of questions.
In the Jungfrau you will tend to set a base and make day trips from it. Hut-to-Hut hiking is a fairly strenuous exercise. You tend to go down into valleys between climbs up to the huts. And, as Lucas noted, they may not be accessible in June. There are plenty of inns and a wide range of hotels/inns/guesthouses/hostels available at various price points:
<<"We love the idea of leaving from a town and hiking hut to hut for our 4
night journey---but maybe it would be way easier to have a home base? We
just think it will be more fun to have to hike to our next place to
sleep, but will change our vision if it just doesn't make sense.">>
Do you regularly hike at high altitudes...say 2000 meters? A mile is approximately 1609 meters, in case you are from the USA.