Doing both is possible, probably not advisable. A rule of thumb is that public transportation the farther south one ventures in Italy is not quite as reliable as it is up north, so relying on clockwork transport could throw your jam-packed schedule awry.
Keep in mind that Naples & Pompei are open, but much of the Amalfi Coast establishments (hotels, restaurants) are not, probably depending on when in November you go--more is open earlier in the month. That being said, the A.C. is out-and-out spectacular just to see, possibly made better by few tourists to have to deal with. On the other side of the peninsula, Sorrento's stuff is open year around, except for some of the places down by the marina.
OK, let''s try a jam-packed schedule for you, 10 nights worth, since that's what you're used to, doing some of everything. Fly into Naples, out of Palermo. A caveat--I would not do this, nor would I try it at home!
Night 1-4: Sorrento. Best base around, best breadth of cuisine as well. Daytrips to Naples, the A.C., Pompei, Capri unless waters rough.
Night 5-7: Taormina. Air Naples/Catania most efficient, train Sorrento/Taormina would take 8-10 hours, and 3 changes. Daytrips to Etna, medieval towns, Giardini Naxos.
Night 8-10: Palermo or Cefalu. Daytrips to wineries, Madonie Nat'l Forest south of the coast.
You could spread yourselves thinner and see more of Sicily, but I'd rather limit the bases and do daytrips. On second thought, maybe I would try this!
Enjoy your planning!