Doing a diff oil change on my 4.0L Auto x300 and about to refill. Should there be a washer on the drain bug?

There was a lot of gunk when draining and the old washer (if there was one) has dissolved into an oily slime.

 

You might have answered your own question. If there was a lot of oily gunk at the drain plug, and you couldn’t find a washer………
Even in the bad old days, which the X300 had no part in, washers didn’t dissolve. On a positive note, it probably means your diff has had an oil change before.