Don’t know the vehicle or even if this is a help or not but according to this the ford 150 and the Lincoln Mk LT share the same service manual.
Looking at the wiring diagrams , (scroll down to wiring diagrams > system wiring diagrams and click on the link - see p.62 of 127 to find schematic) you could try removing fuse #21 as it seems to provide the power to the instrument cluster - vbatt pin 9. I don’t know what happens to any stored settings etc, radio? if the fuse is removed as I think this may provide the backup power for them as well. Check what it’s called in the fuse panel perhaps.
Also there are a few relays in the cluster so if removing the fuse stops the noise on idle then it may be a relay chattering. The trick may be to find out which one.
Just some ideas.