Flying out of BCN late morning. Looking for an Airport hotel that may have a pool or close to beach. Would consider in town hotel as well, maybe not in historic center but with good nearby restaurants for dinner. Would any of these hotels have a free shuttle to or from the airport. Flying in from Andalucia and out the next morning to USA so we need to get to hotel from airport and back to hotel next morning. Any suggestions?

I have stayed at the Salles Hotel a few times - 10 minutes from the airport and they have a free shuttle.
https://www.hotelciutatdelprat.com/

They have a pool, but I have never used it. There are some restaurants around, bars, cafes, stores.. enough that you won't go out of your mind with boredom. Just make sure to bring the shuttle instructions so you know where it stops.

I also stayed at that hotel. Pool was a little cool.

I'd stay in the city and would enjoy the city.

The beaches in Barcelona aren't dreamy tropical destinations. IMHO unless you're desparate for beach time spend your day/night visiting the city. If you really want a beach stop do it in Andalucía which includes some beaches well worth the effort.

You can take a cab to the airport. You can take the airport bus. You can even take public transit. Unless the flight is leaving really early none of this should be an issue.