Try using another wall adapter and charge cable to be sure, but yes, it does sound like your battery has died. Assuming your warranty has expired, B&N is unlikely to replace it. I have heard stories of them offering discount on upgrades to new devices, so it might be worth taking it to a store or calling.
You can find replacement batteries online, but most shops indicate that replacement will cost nearly as much as a new device. With the current sale price on the HD/+, you might upgrade instead of paying so much to repair the NC.