so it's hinting me toward a problem with the TLC Mux library.
From your scope trace, it look like the oscillation is occurring when any one of 3 of your FETs are turned on, but not when the other 5 are turned on. Can you post a scope trace of the signals on the gate and drain of the FET that is 4 time slots further along from the one you originally posted? That one looks to be the worst.
I presume you have the BLANK inputs of all of the 5940s securely connected either to ground or to an Arduino pin that has been configured as an output by a call to pinMode?
One other thing I spotted is that your mosfets are not intended for logic-level gate drive. The data sheet suggests that with 4.5v to 5v gate drive they will probably switch on sufficiently if the current they have to switch is no more than 1A - which should be more than enough for 8 LEDs. However, it will vary between devices. A scope trace should show whether they are switching properly.
So I decided to use some pull the drain down to 0V using 10K ohm resistors
So it was a layout issue after all.
When the FET is switched off the drain is in effect floating which is why you see that slope on the scope. By adding a 10K pull down you provided a path to ground for your scope input, it makes the traces look like you expect them to be, but it is only an expectation. You can remove this resistor and have the circuit work just the same only it doesn't look as pretty on a scope.