The specific cyclic nature of these (a month or so on, a month or so off), and the very long attacks ("hours" is not particularly unusual, but "days" certainly is), suggest to me that it's not CH that you have. When you say "Severity does change slightly during that period but it's very painful the entire time," it does make me think of a condition that is a kind of CH "look-alike," hemicrania. There are different forms of hemicrania (hemicrania continua (https://my.clevelandclinic.org/health/diseases/21538-hemicrania-continua) and paroxysmal hemicrania (Paroxysmal Hemicrania: Causes, Symptoms & Treatment). Neither one is precisely what you have described, but you might be close to episodic hemicrania continua.
Hemicrania typically (but not always) does not respond to sumatriptan injections, which you might have been prescribed for a migraine diagnosis. Hemicrania also typically (but not always) does not respond to oxygen, which would normally be a prescription for a cluster headache diagnosis. So if your attacks have responded to either/both of those, it would cast doubt (but not conclusively) on a hemicrania diagnosis. It's relatively easy to test for hemicranias, since they do respond to a drug called Indomethacin (when it is properly prescribed).
Can you say something about what you have taken to treat your condition? You might want to consider starting on the vitamin D3 regimen, which has been an effective preventive for many severe headache conditions. https://clusterbusters.org/forums/topic/1308-d3-regimen/