While you're in Korcula you should definitely do the walk up Sveti Ilija. It can be reached from Orebic, just across the channel from Korcula and connected by regular passenger boats in summer. The trail is well signposted from the Franciscan Monastery above Orebic. At 961m it's not quite as intimidating as Biokovo - which I've never tackled, there is something about that mountain that I find scary just to look at! But of course starting from sea level it's not to be taken lightly. There is a little bit of shade on parts of the trail, but as almost everywhere on the Croatian Adriatic coast, there are long sections very exposed to the sun (and to sun reflecting off the exposed limestone). Once you get to the top the views are truly awe-inspiring - because the Peljesac peninsula extends so far out from the mainland there is something interesting in every direction.
I don't know where else you are thinking of going in Dalmatia. The islands of Lopud and Sipan are good for gentler walking if you happen to be in that region, as is Mljet, which has more shade than almost any other island. There's also a good walk up Vidova Gora in Brac (although again quite exposed to the sun, but shorter so you could easily avoid the hottest hours), and some marked trails around Komiza on Vis Island.
If you can get hold of the Sunflower Guide to Croatia by Sandra Bardwell, it has some good suggestions for walks of various lengths.