You say it works well with a battery, but spits with an adapter. Is the adapter regulated? That's what I would try first. If you don't have a regulated adapter, it's easy to use a cheap, unregulated 12V DC adapter, a 7809 or 78L09, and a few caps to get a steady, noise-free, regulated 9V. Let me know if you need more guidance with that.
Other than that, I recommend asking your question on
DIYSB. The designer of the circuit is very active over there and will probably be glad to help.