Hi Martin,
Property Transfer Tax in the Valencian autonomous region is 10%, then you need to add 3-4% for fees, e.g. lawyer, notary, land registry. So, realistically, your maximum budget for buying is around €52k.
I've just done a search on www.kyero.com and these are the results:
Although the search returned 98 properties, the majority are in remote Spanish villages. I think the only ones that may fit your criteria are 2 in Torrevieja, 2 in La Marina, 1 in Algorfa and 1 in Formentera del Segura. If you look at apartments or properties in a "community" you need to be aware of annual fees for maintenance of communal areas, including pools.
I think there will be more properties than this available in areas that may be of interest to you, but estate agents have to pay to list property on internet portals and may not believe that's necessary (or affordable) for the less expensive properties. But Kyero does include the listing agent, so you can check out individual agents' websites that may have other properties that they haven't listed on Kyero. From what you've said about your requirements, I agree with GrannyRose that Torrevieja is likely to meet all your needs.
One other thing to be aware of, whilst it is possible to get a "bargain", you do need to be aware of Spain's Complementary Tax:
As you start to identify properties that are of interest to you, check back with us, as there may well be people who have good knowledge not just of areas, but specific communities that you're considering.
Happy hunting!
Kind regards,
Kim