Based on your description I really think you have coddling moth. You do not tend to see a lot from the outside other than a very small entrance hole. The small larvae go straight for the core, eat, molt, get larger,  etc..., then exit via the blossom end leaving a larger hole. The core rots and so does the exit hole at the blossom end. There are variations but it explains a lot of what you are seeing. Have a read here:
http://insect.pnwhandbooks.org/tree-fruit/apple/apple-codling-moth,

Thank you so much for your prompt reply. I looked up coddling moth because someone else told me that is what it might be. The pictures at your link do not look like my apples inside. They are fine looking...I have eaten quite a few. If you cut them in half, the core is black and rotten and of a different texture than what you would expect them to be. But the flesh of the apple is fine, juicy and tasty. That's why I questioned if apples can even get blossom rot? I'm new at the fruit growing game and am about to give up with so many years of bad apples. I have attached picture of 2 apples I just went out and picked up off the ground. Most that I have cut have been totally black at the core...these 2 just have it starting. Hope that helps some. Again, thank you for your help.

I sent this message on to our post harvest expert in Hood River Oregon - Dr. Yan Wang and here is his response:

"I couldn’t see any rot or problem from this attached picture. The apple core in the picture seems healthy to me. Based on the information described in the email, it seems moldy core disease. Moldy core can be caused by many fungi but especially Alternaria and the infections can occur during late bloom and pit fall. It is common, but especially for the varieties having open sinuses at the calyx end, such as Gravenstien and Delicious. The susceptible cultivars should be avoided in the areas prone to moldy core problem, such as with a wet weather during bloomtime. Fungicides and culture practices such as tree training and pruning to help fast drying of plant surface may reduce the potential for spore development and therefore should help to reduce the incidence of this disease."

I do note the very open sinus on the first image you sent. Going after scab will help in the long run. If you are already doing that, then you seem to have a clone that is going to be very sensitive to this problem.
You can also send in samples of the problem to the Plant Clinic which can culture the fungi involved. That might help point out how to manage the problem. (There is a $75 fee for clinic samples.)