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?