Dont have LED's in my rig, but use them at work all the time. I have found many of them are fine, just need a little adjustment.
Some LED's have a two pronged connector that inserts into the housing where the power feeds the fixture. This connector can "slide" out of the housing just enough to do exactly what you described.
Anyway might be worth a shot before digging into the wiring.
Take the cover off the light to expose the led and see if yours is similar. If so give it a gentle shove toward the housing and see if that does it.