There's a lot going on here. First of all, it's hard to move somewhere knew, even if you WANT to go, when you are leaving things and people you hold dear behind. Nothing can replace that. At least it hasn't in my experience.
Second, you have the responsibility of two new little ones. One is hard enough, but two? My two youngest weren't even twins, they were just a year apart, and the thought of trying to get them and all their gear out of the house and in the car made me break out in hives. We didn't have a good double stroller. Ours was huge and awkward. So navigating stairs and stuff? Uh-uh. Wouldn't have happened. I was always afraid that while I was dealing with one, the other would fall, get hurt, run into traffic. Maybe I just wasn't very good at it. Maybe my subconscious brain knew it.
Third, you got really, really sick. If someone hasn't had mono, it's hard to explain the level and quality of fatigue. It's not just being tired. It's being so tired that it literally hurts to move body parts. From what I understand, fibro flareups are much the same. Plus the pain. Plus TMJ. So you have chronic illness. Sometimes people don't understand that there is no "getting over it". You aren't depressed. You are sick. Chronic illness can cause depression, but that's not what we are talking about here.
Yes, she probably did feel rebuffed, because she probably really doesn't understand. Yes, her feelings were probably hurt, but that wasn't your intent and she sounds like she wasn't looking at the whole picture. A little empathy on her part would have made for a lot less heartache on both sides.
As to how you navigate it, the main question is: Do you want to be friends with her or are you just smarting from what feels like rejection? If you weren't feeling left out, would you be seeking her friendship? That's not a chastisement. It's a real question. Sometimes when we feel rejected we want to seek acceptance from that person - kind of make them take it back. Would you be friends with her or seek her friendship if this weren't the case? If not, then Gamma G had some great suggestions on where to find some like minded people who share your experiences and could make for an EXCELLENT support system. I suggest getting out when you can and finding mom groups, support groups and the like. If you would be here friend in other circumstances, then call her and make a time to meet with her without the kids and really open up about what's been going on. Tell her that you are sorry if you ever made her feel rejected. Tell her you should have been more forthcoming about what was going on, but that you weren't used to feeling so helpless and didn't want to be one of those people who complained all the time about what was wrong. Tell her that you would really like if you could start over, that you still struggle, but that when you are, you will let her know so that she understands that it's about your illness, not about her offer of friendship. Then let her make her choices.
Hope this helps. I will be praying for your health and well being and that you soon make a ton of knew friends. It will never be quite like what you left in the city, but it can be a pretty awesome close second.