Health insurance advice--what an enigma

Hello all!

I moved to Hamburg 2 months ago from the U.S. and got my CELTA certificate. I got a job, but now I need to get my residency permit and freelance visa. Unfortunately, I don't yet have health insurance. Yes, I now know it's mandatory by law, and I now know I should have taken care of this issue upon arrival. However, now I'm in this situation and I'd like some advice. It looks like I have to get private insurance, but I've also been reading that many companies will not accept non-EU citizens. I've further read that I may be required to pay for insurance retroactively for the time that I've been here without paying. And lastly, I've read that there's a bit of a catch-22 in the system in that in order to get insurance, one must show her working visa, and yet in order to get a working visa, one must show proof of insurance. Does anyone have advice for me? Where to look for insurance in Hamburg and what companies will be willing to cover me? I'm a 27 year old female, single, non-smoking. Any advice is greatly appreciated!

Thanks!