Reducing noise in mixed-signal audio circuit
The core problem that you're experiencing is probably that the SID is not a low-noise device.  When I had a C64 way back in the day we would turn up the volume to listen to the noise-- as a debugging aid.  We could tell, roughly, if the CPU crashed by the noise generated from the SID!

That being said, there are some things that you can do to minimize the noise.  I have no idea if any of these will reduce the noise enough in your case, but it's a start.  Here goes, in no particular order:


Use some linear regulators to give the SID its own power rails.  Use low-noise regulators.  Use large ceramic caps (make sure the regulator can handle them) instead of electrolytic or tantalum.  You might only be able to get these caps in SMT.
Filter the output of the SID to remove high-frequency noise.  A simple low pass filter set with the cutoff at around 16-18 KHz would work.  The idea here is to reduce the audio that you can't hear so that it doesn't interfere with downstream devices.  (Someone is going to say that they can hear up to 20 KHz, ignore them.)
Use series termination on all signals between the CPU and the SID.  Use the highest value that you can get away with (start with 50 ohm).  This should reduce the edge rates and overshoot/undershoot on the bus, and reduce that getting into the audio.


There you go!