Based on the description, I'd guess a light leak. I've never heard of chemical exhaustion causing black spots; the more common result is low-contrast, light, or mottled prints.
Are you storing your paper in the box and bag, in a paper safe, or in some other way? Is your paper stored emulsion side up or down? Do you turn the lights on in the darkroom during your session (to evaluate prints or switch negatives), or do you stay in safelight-only mode the whole time?
I've seen this sort of thing myself, but only when the paper has been stored for a long period (weeks or months) emulsion side up in a paper safe, or once with paper stored in a box and bag in my freezer (I'm guessing there was a pinhole in the bag).