I'm out of touch with the best places to stay in Cape May, having not lived there for sixteen years, but don't let anyone tell you that November isn't the best time to visit. It can be spectacular. Any cold fronts in late October through to about the second week of November can deliver massive falls of Robins, Sparrows, Yellow-rumped Warblers, along with hundreds of Hawks, Gulls, Goldfinches and dozens of Hermit Thrushes, Bluebirds, Meadowlarks, Woodcock and so on. It is also a month local birders find more western and south-western rarities than any other time of year. I've seen Brown-chested Martin and MacGillivray's Warbler on the same day there as well as Western Tanager, Western Kingbird, Ash-throated Flycatcher, loads of Cave Swallows on separate occasions... oh, and Owls: it's good for Saw-whets at that time too, although you'll need to enlist the help of the Bird Observatory to assist you finding them.
Please note, the above is a distillation of about five Novembers spent there. You definitely won't see all the birds I mentioned, but keep your fingers crossed for - I'll emphasize it again - cold fronts, and if it happens, you're laughing.
Good luck