dc42:
OK, what you need is serious decoupling on the LED modules. On each one, connect a 100uF or greater electrolytic in parallel with 0.1uF or greater ceramic between the +ve feed to the LEDs and the ground pin of the TLC5947. Preferably, also connect an inductor between the incoming LED power feed and the junction of those capacitors with the +ve side of the LEDs. As an alternative to an inductor, you could use a low value resistor, if you can afford the voltage drop it will introduce. The idea is to get the LED switching current circulating around the capacitor, instead of feeding down the power and ground lines.
How would I go about selecting an inductor for this purpose?
Also, do you have any suggestions as to how I could isolate the audio portion of my circuit better? The noise from the LED modules when running the amp on it's own power source is nothing compared to the noise generated when I connect servos to my circuit without powering them from their own battery. My board is very small and I can't go attaching huge capacitors to every IO port. Is there some combination of voltage regulator, capacitors, and inductors that I could put in the audio circuit that would minimize the noise from the servos, and presumably take cake of the noise from the LEDs as well?
I don't think a voltage regulator alone would cut it because I've read they don't typically reject noise over 100hz, and the ground would still be connected and I suspect noise would get in there.