apple fruit diseases
What your apples have, as you suspected, is core rot and not moldy core. There often is confusion about moldy core symptoms. Moldy core is restricted to the core; it does not progress into the fruit flesh. When the fungi spread from the core into the flesh, causing the flesh to decay, it is called core rot. Moldy core and core rot are two different diseases. Fungi that are found in the core can be from several fungal species, such as Penicillium, Alternaria, and Botrytis. Depending on the infecting fungi, some moldy core can turn into core rot, but this does not happen all of the time. If you encounter an eating apple that has moldy core and not core rot, the flesh is still eatable should you choose to eat around the core.
In some studies, fungicides used during bloom have been successful in controlling moldy core, but the results are erratic. However, if you would like to try a fungicide use one for scab and powdery mildew during bloom. Also recommended are cultural practices, such as removing leaves from below the tree canopy, tree training and pruning (which open up the tree canopy and allow good airflow and light penetration), will help to promote fast drying of plant surfaces, reducing the potential for spore development. These practices should help to reduce the incidence of moldy core.
Good luck and happy gardening.