I've never overnighted at Little Dix nor Savannah....might be a little rolly in winter months (ground swell). You could try Cooper Is (Manchioneel Bay), but that usually fills up early and they don't really want you anchoring there. They do have boaty ball moorings there if you want to try and reserve. Its 6.6nm from Savannah to Scrub Is marina and 5.4 from Cooper....so about an hour sail either way. Another option is Marina Cay, which is right next to Scrub and a very nice anchorage with plenty of mooring balls if that's your preference. Very good snorkling at Diamond Reef as well. If I were you, I'd try Cooper first and if that didn't work out, head on across to Marina Cay.

Good luck.