Here is my experience-- and keep in mind that (presumably) many, many women who don't have any issues don't often post here :-)
I am 36, history of depression, and never tested for any of your issues like anemia or low thyroid/copper. But I had the mood swings on every pill and patch, as well as more or less constant breakthrough bleeding. I have been pregnant twice and quit Paxil cold turkey and the mood swings on birth control were worse than either.
Given that, I tried the Mirena IUD between baby 2 and TTC (now) with hopes that it wouldn't have those issues, and it was estrogen-free.
The insertion did hurt a little (should have taken Advil first) but I know it hurts more for women with no children. I did have 6 weeks of minor irregular bleeding and cramps, which I was warned about. No biggie. My issue was that after 6 months, I still had spotting and cramps-- not BAD cramps, but then I didn't think most of my labor pains were that bad. And some nausea. I can't really comment on sex drive because I had an infant I was nursing, which also affects that.
The mood swings I had with the IUD weren't as bad as with the pills. Not nearly! But they were still there. My feeling was that I had minor PMS, but since I spotted 2 to 4 times a month (no real period-- just spotting! Which is an improvement over pills also), I had PMS each time-- and it was unpredictable. So it was a real annoyance. Nobody believed it could be the IUD, but it's been out 6 weeks with instant major improvements in all symptoms, so I think it was.
Given how bad your situation is now, the only negative I'd seriously consider about trying the IUD (if your insurance covers it) is that it can take 2-6 months to get back to normal if you do have an issue.
Personally, I am done with any hormones but it's your choice. If you do go ahead with the IUD, please ask your doctor if there will be any problems/costs associated with its removal, and how long he'd like you to try it before you decide. I'd also ask if the Mirena IUD is not contraindicated in people with a history of thyroid or depression/anxiety issues, and if your doctors think the benefits outweigh the risks (this will also ensure they know all the risks and possible side-effects, and help you monitor for them)
I wish you the very best and a speedy, definite improvement in your condition!