It won't sync to the software but you can still sideload the books. In your HD's menu, under account settings you can set your Adobe ID. Then you can just sideload the books from your ADE download folder on your computer. The ADE software does not support the HD (and other newer tablets) MTP connection protocol, so the hardware workaround above should do the trick.
Also, you can download the free Overdrive app and download your library books directly to your HD. The difference here being that you are using Overdrive reading software vs standard nook reader.
As previously stated, the ADE software doesn't support the newer tablets. You need to activate ADE on your Nook HD directly. And as previously stated, you can download the Overdrive app aud use it to download library books directly to your Nook HD and read using the overdrive App.
If you'd rather read library books using the Nook reader, what worked for me was this.
Activate ADE on my Nook HD in the account settings.
Check out and download the library book to your PC.
Open in ADE. You have to do this because it's .ascm file and ADE actually downloads the actual book.
Connect the Nook HD to the computer.
Using Windows Explorer (not ADE, since ADE won't recognize the Nook HD), copy the library book to the Nook HD files section.
Go to the files section on the Nook HD and find the library book. I was able to open and read it that way.
Hope it works the same way for you. However, it's easier to read it in the Overdrive App.
Someone has also stated that one of the other apps, the Alkido app maybe, will also connect to your library and you can download library books directly to it. I haven't tried it and I'm not sure if that's the app, but someone posted about it, so you might be able to search and find it.
To side load te library books onto the Nook HD and use the native reader takes a few steps, but is far and away better than using the ADE app.
Download the book from the library onto the computer, then place it in ADE, the book will now show up in your digital edtions file.
With the Nook plugged into the computer, open computer, and you will see it listed as a digital media device. From the digital editions folder, drag and drop the book to the Nook
Open the Nook on the computer and find the ADE file. right click and delete it. With the nook still plugged into the computer and connected to WiFi, reboot the Nook.
After the reboot, go to settings and log back into ADE...
The book can now be read in the native reader...