Welcome to the pack.
My previous beastie had a dandruff problem that was brought on by stress – but it wasn't nearly as bad as you describe. The oil supplement has been suggested before and may solve your problem too. Make one change at a time though so you'll find out what really works.
And definitely post pictures when you can - we are just addicted to them (you will be too).
If you have any photos of your Basenji(s) It would be great if you can post them in the Show Off Your Dog thread. Please also browse through the introductions of other new members and say hi.
Let us know if you need help posting your photos or have any other questions about getting around these forums.