You're probably right about the wires. There are wires going from the PIC to the LED display pin header sockets, and to the 3 column driver circuits. When I made the shields, I put them over the EMF amp components and tucked them in so the EMF bits are inside the shield and the digital circuits and display wires are outside.
With the LED display connected, I get noise that varies depending on which/how many LEDs are lit. If I disconnect the display, then I just get a continuous buzz noise, which is most likely from the column drivers, which are on the main board to the left of the EMF op-amp stage. If I disable the display completely within the firmware (so no LEDs are on and the multiplex pins aren't being strobed) but leave the PIC running, I get no discernible noise.
The EMF circuit's ground is connected at one point IIRC. I'll have to double-check this, but I don't think it's that anyway. When I was experimenting with the bypass caps, I had a scope connected to the EMF power, and was getting voltage fluctuations corresponding to the LED current draw. I added the 250µF and still get fluctuations, but they're smoother, and no change in the noise. When I stuck a 1000µF in parallel, the power fluctuations went away completely on the scope, but the noise didn't go away or change at all. (This was before I added the shielding though).
I'll try the bigger cap again to see if it makes a difference. I thought of the resistor, and instead I put a diode before the cap to see if it would have any effect (I have other reasons for putting a diode in there which I can explain later if you're curious).
FYI, the circuit is powered off a 9V battery. Power is switched on/off through a MOSFET (allowing the power to be cut via the firmware using a button), the EMF circuit gets unregulated power (battery voltage less 1 diode and any drop in the MOSFET), and the digital bits (PIC, LED stuff and temp sensor) are powered through a 78L05.
Maybe I'll make some additional shielding to put on the wires going from the PIC to the LED bits.