If it is migraine, you would think that the rizatriptan (Maxalt) would help, since that's what it's for. It NOT a cluster headache medication (or at least not a first-line CH med). Not saying it isn't migraine, of course, just observing.
I will jump to my usual possible diagnosis (NOT a doctor) of some form of hemicrania. The eye droopiness and watering are symptoms of hemicrania, and the continual pain with occasional explosions could fit, too. Hemicranias are typically resistant to triptans (although it's possible that the "painkilling shot" that helped you was a triptan, and as I've said, rizatriptan pills typically don't help CH in any event, so that isn't much of a test). Here is some information about one form of hemicrania: https://www.webmd.com/migraines-headaches/hemicrania-continua-symptoms-treatment Another is paroxysmal hemicrania, which you can look up. A "good" thing about hemicrania is that there is a highly reliable preventive, Indomethacin.
There isn't enough info right now to do anything but guess and be sure all the options are being considered. I am just providing this information so you can be sure you and your doctor is considering/aware of all the options. For all I know, it could in fact be sinus-related and/or Covid-related.
You might want to look over this file, just in case -- https://clusterbusters.org/forums/topic/6213-basic-non-busting-information/ -- and maybe you'd want to try some strong coffee or an "energy shot" such as 5-Hour Energy, since those typically help reduce CH pain.
Doctor oddities appear much too often: In this case, diagnosing CH (that's a good thing, if she is correct) and then prescribing a migraine medication that is explicitly not for cluster headaches (https://www.mayoclinic.org/drugs-supplements/rizatriptan-oral-route/description/drg-20065868#:~:text=Rizatriptan is used to treat,group of medicines called triptans.). If this painful situation continues for you (I hope/we all hope it doesn't), you're likely to get the best diagnosis and treatment at a headache center.