3 months rent in Puglia

I don't know anything about property rental in Europe, but I want to be sure you realize that US/Canadian citizens are limited to a maximum of 90 days within any 180-day period in the Schengen Zone. Italy is part of the Schengen Zone, so unless you want to go through the hassle of obtaining a long-stay visa, you cannot stay in Puglia for 3 full months. Count your days very, very carefully, and remember that both the arrival day and the departure day count. Overstaying the Schengen limit is likely to result in a substantial fine and being barred from re-entry for a number of years.

Thanks for the information re Visas.....it is the same for us Ozzies too.
We are planning to rent for 12 weeks to allow for delays etc.

Good. It was an Australian couple I met in Italy who explained the details of the rules to me back in 2015. If not for them, I would have overstayed the limit and gotten into trouble when I headed out of Slovenia. I am everlastingly grateful to them.

You HAVE to be in Bari for the Festival of St Nicholas, around 9th May. The most amazing street parade, drummers, horses, a woman doing aerobics while supported by a bunch of helium balloons.
The Italian air force helped out too, an aerobatic display opened by an aircraft flying inverted over the harbour at an altitude of what seemed like ten metres. Don’t miss it!

I would check VRBO, Homeaway, or holidaylettings from Tripadvisor. That would give you a good idea of what is out there. I saw properties from Late April for 88 nights for $5,000 to $20000.
Looks cool. May have to check that out myself.
I have travelled since the mid 90's and rarely have I stayed in a hotel.

Have used all of the above rental sites as well as Air BnB and local vacation rental sites.

Look at Polignano a Mare and Monopoli, beautiful waterfront towns south of Bari.

If you can avoid August and maybe the latter part of July. It’s harder to find vacancies then. Anything before mid July is not crowded. I prefer spring to early summer, rather than autumn because days are longer. Temperatures are also more amenable, at least before July.

Polignano a Mare is a good choice.