@Brandon296 How frustrating and yes, ludicrous that AirBNB isn't making this right for you without a hassle. It isn't surprise, though. Their customer service has been tanking by leaps and bounds lately.
Good of you not to blame the host. As far as her declining to refund, what you may not realize, as a guest, is that the host actually doesn't get paid for a reservation until 24 hours after the guest checks in- Airbnb has your money at this point, not the host. So the host has not received anything to date, either from your booking or the one she got when your booking was cancelled. Many hosts are leery to approve a refund, because sometimes a host offers to refund, then the guest demands a refund from Airbnb and gets it, so the host has lost out twice. I know this is hard to believe, but as you've found out, many things Airbnb are ludicrous and unfair.
Perhaps if you message the host, letting her know that you are aware that that at this point she has received no money to refund, but that when she does get paid out, if she would be so kind as to refund since she did manage to rebook the dates, she will be more amenable. And she may not be aware at all, unless you have told her, of the reason your booking got cancelled- she may just think you changed your mind, so is sticking to her cancellation policy. Airbnb is very tight-lipped about disclosing anything about the guest's info or payments to hosts.
Keep at Airbnb, though. Persistence seems to be key with them. Also ask if they really want to lose a guest from their platform over this, that there are other booking sites out there that you will use in the future if this doesn't get resolved. But try to stay nice and polite with the CS rep, even if you feel like screaming at them. They don't make the rules, they are just lowly employees being bombrded with angry hosts and guest complaints all day. You'll get better action from them if you stick to your guns, be firm about what you need them to do, but not take it out on them personally.