There's a few options available. Usually the villa owners will store luggage for you to collect later but is this isn't a possibility you could book just one night or day rate for a room in the cheapest hotel you can find, maybe start looking now. I know people on FOK rave on about the Elixir but we had a bad experience there due to them trying to charge us for our girls to wait for us while we spent a good part of £100.00 on a massage there. So personally I wouldn't recommend.
We've had lovely days at the Asfiya, no charge to use facilities but obvs need to buy lunch drinks etc. We've never asked for a room though. Just gone during the day.
Indigo beach club could be another option as they have changing facilities, also you could consider a boat trip and take your airport clothes to change into? Limited space but they do have toilet and washing facilities I suppose. I think Indigo can also store your luggage too and have showers/changing rooms.
I've also heard that the Kelebek hotel is reasonable to a day room rate but not personally had experience.
Hope you get sorted.