I'd recommend going to Nice and staying at Novotel Nice Centre.
Sunshine is almost guaranteed but it's not too hot and it’s only a 2hour flight. The promenade stretches along the sea front for 8km and is perfect for pushing a pushchair. There are lots of gardens and green spaces and it's always easy to find a bench or a shady spot. Great for feeding your LO or just people watching!
The hotel has baby baths and baby sitting on request and family rooms are generously sized. It has a rooftop pool with sea views. Children get a little welcome gift and it's worth noting that there's a children's library and play area. The hotel is a few minutes from the tram station and Nice airport is just 9km away. www.novotelnice.com/en/the-hotel/services-facilities

Barcelona

August is REALLY hot in the Mediterranean and I'd avoid it with such a small baby. Why are you going in August - are you teachers? If not, and there is no particular reason, I'd wait until Sept. It's cooler, cheaper and much less busy.

As to where, when our DC were little we aimed for pleasantly warm, rather than roasting hot as babies can't regulate their heat like older DC/adults can. We had a good holiday at Emerald Coast Gites in Brittany, a lovely week in a SC cottage in Lyme Regis, Dorset, La Grange du Moulin in Charente Maritime. Menorca and Majorca are favourites too with lots of families - Pollensa is always talked about favourably - but I wouldn't go in August!

We spent most of the summer in the Algarve when my daughter was six months old, it is hot but manageable. I would say a villa with lots of shady areas would be better than a hotel, we used to go down to the pool about 8.30ish for a couple of hours (our pool has a big shady veranda next to it) or go to the beach whilst it was still relatively cool, have an early lunch and then chill in the house or out on the grass (in the shade) until late afternoon or maybe go out in the car for a bit/go for a wander round town in the shady streets and then go back to the pool for an hour or so about 5ish. We took quite a lot of toys with us to keep her entertained and it was great, we also took it in turns to do other things, so each afternoon we had a bit of time to sit and read or go for a swim whilst the other parent watched her.