Just to be clear, I'm talking about the new mysqli extension.
That said, I did this to get the new mysqli extension running:
1. Removed MySQLi Server Behaviors 1.0.0
2. Launched Dreamweaver CC 2014
3. Clicked on Windows > Browse Add-ons... to go to the Adobe Add-ons page
4. Quit Dreamweaver
5. Found Server Behavior & Database add-on quickly by viewing just free add-ons.
6. Installed this add-on from the Adobe site.
7. Launched Extension Manager CC and installed WebAssist's MySQLi Server Behaviors.
8. Quit Extension Manager and relaunched Dreamweaver CC 2014.
Now the extension loaded from webassist and all seemed well.
The downshot was that I assumed that the WA extension stood on its own and didn't require at least the previous existence of the Server Behavior & Database extension (deprecated), but that apparently was not the case.