The first thing that you absolutely need to do is to get a better multimeter. The reading of 300mV of noise indicates that your meter can't resolve low AC voltages, so even if you could break the circuit down to isolate the noise, you can't measure it. If at all possible, take that one back to Radio Shack (FYI - the last six meters that I bought at RS were all defective)
Try the $25-ish multimeter from Harbor Freight, or borrow a different meter from a friend, then recheck the noise at the speaker posts/headphone jack. A very loud amount of noise in your application might be something like 10-30mV.
-PB