Can I get a bus from La Azohia to nearby towns.

You really do need a car. There is a bus service to Cartegena but I don't know how regular it is. La Azohia is even quite a drive to the Port. I rented there for a month once and decided it was too far to drive to the Port on a regular basis.

Get a bus timetable from the tourist information centre in the port, you can get a bus in La Azohia at several stops we use the service to go to Cartagena for the day and leave our car at home so we can have a drink with a meal.

The last bus back from Cartagena is about 2120 hrs 

We stayed there last year there is or was a bus goes to La Azohia all the way the end then turns around to go to Mazzaron ch eco time table

There is a bus schedule from La Ahoia through Isla Plana to Puerta de Mazarron