North Carolina has been under a state of Emergency since Tuesday. Hurricane Matthew has been all over the news for over a week - especially in FL. Deciding at the last minute to change their trip was their decision and they should cancel their reservation. Do not cancel for them. Do not accept any emails from Airbnb asking if it's okay to cancel (because you are not okay with it and do not want to be penalized). Send them this information: Here is how you cancel your reservation: https://www.airbnb.com/help/article/169/cancel-a-reservation-as-a-guest
Assuming all your listings have the same cancellation policy, your moderate policy says you are entitled to some of the money if they cancel. This policy exists exactly for the reason you are describing - that you have an opportunity cost that adversely affects you when a guest cancels last minute.
Now you should encourage them to ask Airbnb for a refund based on Extenuating Circumstances. You do not have any say in whether the guest can receive a full refund under that policy - that is up to Airbnb.
Personally, I would say no because you are already into the weekend. They waited too long too make this choice. Even a week ago, when everyone knew Matthew was coming, they should have made a decision. Of course, if Airbnb says it's okay, well, you won't get paid anyway from what I can tell (there is a discussion about this elsewhere going on right now).
BTW, this is exactly what Travel Insurance is for - to cover you in the case of an unforseen situation. My guess is that they didn't buy any for this trip. That doesn't mean you have to pay for their decision not to chance the roads (which, btw, are probably totally okay - everything here in Durham is just fine too).
There's an old phrase, "Failure to plan on your part does not constitute and emergency on my part." and I think that sort of applies here.