I like Nerja, and it is a beach town.

Achieve total touristic awesomeness and head for Ronda..

www.andalucia.com/ronda/
Information about the town of Ronda, one of the loveliest and most historical town of Andalucia, southern Spain.

Another Ronda vote. Fabulous geology.

www.fincalospastores.com

my fav lodging beautiful even

Andalucian Horses de los Conquitstadores...

Ronda is larger than Cordoba, but it definitely has a small-town feel and is very family/child friendly (I recall the square in evening with children playing while parents and grandparents sit in cafes or stroll). It is also well-served by the bus and train. Of course there are many more much smaller towns if that is more what you are after.
For beaches, I would look west of Cadiz--around Vejer de la Frontera. I did a day trip to Vejer and saw gorgeous beaches from afar.

'I want a town like the English villages in the Cotswold' - Not possible unless you explain further what this means to you.
'A few' in these places - have you time left available?
'I don't want to rent a car - prefer to use public transportation unless a car is really necessary.' - perfectly feasible, but at what stage - the whole country or this last stage on your wish list?
'We travel on a budget, usually staying in 3 star hotels and prefer breakfast included.' - Is there a question?

Thank you for your responses. I visited Ronda many years ago but just for a day--that is a great suggestion. I will look into beaches west of Cadiz. Last time we were there we enjoyed the beaches in Torremolinas but we wanted something different this time.

Another vote for Ronda.

And, there is a great sunrise horseback ride in the gorge that you can do.

Pillow,

I might have helped, but like many posters, you think my questions are rhetorical. Good luck.

Yorkshire, Ronda (pop 37.000) is not bigger than Cordoba (pop 320.000).
Ronda seems to be the place you are looking for. You will not need a car as far as you lodge in one of the many hotels in town. Friends of mine stayed recently at Hotel Ronda www.hotelronda.net/ and were pleased.

Thank you, josele. I'll suggest to my nieces that we spend a few days in Ronda this visit to Spain.

WomBatt, I ignored your comments--too much like my English professor husband. I wanted to make my question short with as much information as possible. No car; small town; small budget. Other responders were very helpful. I knew your questions weren't rhetorical--you wanted more clarification than I felt like providing.

Ronda.

Not as small as a Cotswold village but for that you would need a car. We loved Ronda (we had the ultimate tour guide - Hi Josele). Hotel Ronda was great and very reasonably priced. Here are my photos of Ronda (http://www.pbase.com/annforcier/spain__malaga_ronda_gibraltar)