Sounds like a physiological problem -- pictures might help us.

Possibly brown core? Check out:

http://postharvest.tfrec.wsu.edu/marketdiseases/browncore.html

Berafin:

Did you have a really wet growing season last year? Just asking as I did and noticed a lot of unusual happenings with my apples. First, the amount of apples rotting all summer on my trees was way more than usual. I know some varieties suffer from summer fruit rot but despite using Captan every 14 days I still had a lot of summer rot on Haralson & Honeycrisp. My Albemarle Pippen suffered from it too once we got to early September.

Despite a great apple crop last year, I had a terrible Haralson crop. Due to the really wet growing season, my Haralson apples kept splitting all summer long. Worst crop I ever had on them.

What you have sounds like a physiological issue. Yet I wonder if perhaps a very wet growing season plays into this issue?

Thanks for your responses - researching the link offered above, it makes more sense to me that we have a slight boron deficiency and that we DID harvest leaning toward more mature fruit and storing them cold. We do bring some into the kitchen to use, which is of course, much warmer than their crude storage. For spartanapples, we had a very dry growing season in 2017, but that has never been an issue here. There was no indication that these Rome Beauty's were compromised until late December when I was cutting them to make a crisp. I'd shared them right after harvest and they were perfect then, in early November. I'll salvage what I can and make another dandy apple crisp! Better times for 2018!