Exactly what Charlene said about Albergo Lucy in Trani --- we stayed there for several nights in 2015 and could make nice day trips from there.
We did not think much of Vieste or the whole Gargano, but we did go there in November and are not interested in beaches anyway. We drove through the woods and along the coast --- I don't really get going all the way to Italy to see woods. There are just so many more interesting places to go in Puglia (and I'll add nearby Matera).
We loved Ostuni and stayed there for a week, but in an apartment in the old town, so not useful to you. But some of the best food we've had in Puglia in several trips. It was late March, so not heavily touristed then.
What you might like, assuming you have a car, is the agriturismo we always recommend to everybody: Masseria Aprile right outside of Locorotondo. It's a tiny bit farther from the coast than Ostuni, but you can stay in your own trullo and have an excellent breakfast every morning. Lovely hosts.