Hi Mark,
LED's are fast acting and fire at a much lower voltage than a globe, and are generally current limited by resistors within the product housing.
The flicker is almost certainly caused by a voltage spike from the very large solenoid on the auto trans which is activated by the micro-switch on the throttle body when the gas pedal is pressed.
However looking at the wiring diagrams, the turn signal indicators on the dash are wired in parallel with the main turn globes, and those circuits are isolated from the ignition circuit until the turn switch is physically operated, so they should not see the spike unless the indicators are on.
Like Cees, i'm leaning towards a bad ground.
Mark are you aware that each instrument has a brown grounding wire that is connected to the instrument under one of the two thumbscrews which hold the instrument in place.
When working from the outside, it is easy not to notice those ground wires are there ??
I think this might be your problem....