Sorry if I'm telling you stuff you already know. Car engines tend to make a lot of very 'noisy' signals that interfere with electronics - basically what should be nice smooth voltage levels end up being horribly uneven squiggly lines. Car electronics go through pretty rigorous design & testing to take account of it. My post above was wondering whether that's the case, speculating that with a small amount of power, the noise could be enough to interfere, where as with full lights then the power would be much higher, and the same amount of noise would be less in comparison.
Would canvas a bit more opinion to see if it's common or not to have these flickery lights.
After that check if there is anything else that is being switched on and off at the same times (listen for changes in engine note and general noise etc - fans going or something).
Would clean up the earth connection for the lights too.
Finally, I can't comment on how sensitive the led's are to the flickering, especially without knowing how bad the flickering is. Do you know if the new lights are just a string of 12v LED's or is there other circuitry with them? Chances are they could well be fine to swap over anyway.