Re:A common usage of plum (without the 'b') is as an especially desirable position, assignment, or reward

Yes, like:
Little jack horner, who sat in corner,
Eating his christmas pie,
He stuck in his thumb, and pulled out a plum(b)
And said "What a good boy am I!"

(but i have read that nursery rhyme is not that old, and Jack Horner--really was a political appointtee, who helped himself to some lands, in the time of George III.)

and We're plum out of honey' would be just as common in parts of US as in UK.