@cherylpconnolly
Welcome to the Expat.com Forum and good luck with your potential move!
Spain's a great country with lots of nice places to live. There are no significant downsides that I'm aware of, unless you hate sunshine, beaches, paella... and The Spanish. :-)
I think most of it is hot (a possible downside for some), and I'm not sure it's easy to escape even if you head north to Barcelona/Costa Brava area (which is tip-top as @Lmflmf1 suggests). The coolest is maybe on the other coast (Costa Verde, Bilboa/Santander way), or up in the mountains (Pyrenees, Sierra Nevadas, etc.). The best bet is to accept that summers are very hot, and make sure you have good ACs!
We live further south on the Costa Blanca (Alicante region) in the UNESCO-listed city of Elche. This coast is lovely, and Elche/Alicante (and Valencia) are great options. But REALLY hot in summer, so AC is a must. (AC units are pretty typical in hotter countries like Spain, and can be used for both cooling and heating.)
There is a Dual Taxation Agreement between UK and Spain, so you certainly won't be taxed twice on your pensions. I doubt your tax burden would change much. You should feel better off as property prices are lower, property taxes are MUCH lower, and the cost of living is slightly lower.
Yes, UK pensioners can get an S1 for free (or nearly free) public healthcare in any EU country, including Spain. And Spain has a good healthcare system.
The people, food, culture, coastline, fresh produce are all great. I can't think of anything I would complain about. (Well, maybe the heat, as mentioned, or our electricity bill in summer for running all the ACs.)
As pensioners, you should look at getting an NLV (No Lucrativa Visa) which is very popular and fairly easy.
For property, I suggest Idealista, a very large website with listings all over Spain (from owners and agents, for rent and for sale). Even if you don't use it to buy, it's a great research tool to get an idea of what's available in your price range. You might find your budget will go a lot further if you are guided by lower price regions, rather than fixating on a particular location that might be expensive (Barcelona is the 2nd city, and a very desirable location both in and near the city).
We're in Elche (a city we had never even heard of) because of this price differential. We would have loved to be in Playa San Juan, right on the beach in downtown Alicante... but a similar apartment here (about 20km from ours) would be 5x-10x the price. Luckily, Elche is a lovely city and we like it a lot! :-)