Whew, sounds like a lot to deal with. I can understand wanting to preserve your ovary! At some point, you may be very grateful for that. But would wonder why they didn't cease your ovulation while you are healing. I'd have guessed they put you on birth control so you don't ovulate. I'd go that route or talk to them about that as perhaps an easy solution for the time being while you heal.
You had the ovarian cyst removed but can't get any follow-up from the same doctor? Not even one appointment for some advice? Did you have health insurance at the time and have since lost it? Any opportunity to get it back sometime soon?
If your ovarian area was hurting at ovulation time, I'd say that is sounds like the ovary is swelling like ovaries do during ovulation, and the pain is coming from the recently operated-on ovary. But if it is for sure hurting at period time and not at ovulation time, it sounds more like endometriosis. That's when endometrial tissue (from the lining of the uterus) has migrated out to where it doesn't belong, and attaches somewhere in your abdominal cavity. It swells like the inside of the uterus does, and builds up when the period is coming. Migrating endometrial cells could readily have happened during the time when your ovarian cyst was stressing the whole area -- endometriosis is not uncommon, a lot of women get it. If you do have endometriosis, even had the doctor removed your ovary, it might not have made a lot of difference, it's the misplaced uterine tissue that's doing the hurting, not the ovary.
Keep in mind, this is just a guess about what is going on; I'm not a doctor. I hope it's just pains from healing, but if you have endometriosis, and if it continues this painfully, only when you get back on some kind of insurance and can have another operation will you be able to get rid of it for good. And, I hate to say it, but sometimes surgeons are not that good at finding all of the strayed endometrium, so even surgery sometimes doesn't get rid of all of it.
I'd see if you can at least get a phone call with the surgeon who did your cyst removal, and just run this by her to see if the theory makes sense. (That is, if you are sure the pain is with your period and not at a time when you might be ovulating.) At the ER, they would probably only give you painkillers, and one thing to ask the doctor is if the endometrium swelling and presumably sloughing off time after time is going to give you problems in the future, such as dead endometrial cells in the abdomen or anything like that.
I second specialmom's suggestion that going onto the Pill in order to give the area a break is a good idea. For one thing, at least your ovary can heal all the way. For another, if you were to take (for example) three months of active pills end on end in order not to have a period, and then go for a week off and have a period, and then to take another three months in a row, at least you would only have to deal with the pain once in three months instead of once a month. You could run this idea by your surgeon as well.
I hope you are able to get health insurance again soon, it's one of the main things in life in a system where there is no national health care mandate. If you do get it, be wary of any kind that would reject you having the endometriosis operated on because it's a "pre-existing condition." If you are in the U.S. and get insurance due to the Affordable Health Care act (I think it's Healthcare.gov), I believe they can't refuse treatment of this based on the fact that it was already there. But some health-insurance companies will try to. So stay alert.
Anyway, regarding the ER, all they will do is give you painkillers, and I've had ovarian cysts in my life and when they did give me a painkiller, it worked so well I could have kissed the doctor. So that's not fanciful or a stupid thing to take advantage of if it is offered given how much it hurts when it happens. I would be concerned about the risks of opiates, and also, I'd try to find out from the doctor about whether just ignoring endometriosis and blocking the pain, is an OK thing to do long term. Maybe it doesn't matter. It would bug me.