IIRC, the ferries from Kyushu to Shikoku are a little long. I have a figure of circa 5 hrs in my head, but you need to check that because my memory is really old and we never actually did the journey. We'd looked at them on a previous trip but ended up not doing them. Possibly because of journey length, and possibly because we also weren't planning on being on the east coast of Kyushu, so it was easier to return by train. your plans may be different.
We have taken the ferry from Hiroshima to Matsuyama (some years ago, lovely journey across the Inland Sea) and spent a night in Dogo Onsen before heading to Takamatsu by train, visiting the lovely garden there, staying overnight and then going on to Naoshima by ferry.
We did the ferry from Shikoku to Wakayama in April '18 and it was great. We did a night in the Iya Valley (renting a car from Takamatsu - where we'd flown to from Tokyo), sampled the cable car onsen at the Iya Valley Hotel (completely bonkers, in the nicest possible way) and returned the car to Tokushima (also nice enough to spend a couple of hours in) before getting on the ferry to Wakayama.
I can't comment on Shirahama as a place to stay. We were driving from Wakayama to Katsuura and just stopped off in Shirahama for as long as it took to have a bath, grab a quick lunch from a combini, and drive on. My impression of Shirahama was that it was a seaside, resort-y, place with a lot of concrete hotels. It was April, so not really sunbathing weather, although warm-ish, so the beach didn't really appeal. We hadn't planned to stay there, and I didn't see anything to change my mind as we buzzed through.
Katsuura is a good place to go to Nachi Taisha from. We were on an onsen-focused trip, and we'd booked a stay in an old-fashioned place in Yunomine the next night, so we went bubble-economy crazy and stayed at the Urashima. That (and the fact that like you we were heading to the shrine the next day), was the sole reason we stopped there.
Driving from Wakayama, having stopped briefly at Shirahama, and following the fairly quiet and, sometimes very scenic, coast road, we got to Katsuura at about 3 pm IIRC, so we could have pushed on to Yunomine (for example). Katsuura bay is pretty, the seafood at the market is supposed to be good, but other than that it's a modern fishing port/resort place. A good contrast to Yunomine, and some nice coastal views before you head inland. Otherwise, nothing special.
One other possible place for a drive-by (or day trip from Wakayama) that may interest you is Yuasa, where there's a small preserved old streets district and a soy sauce brewery that does a little tour (and sells soy sauce ice cream - which is lovely). There's an NHK World travel program on it, which should be available on demand in the 'net.
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