Thanks, Randy. Every time you reply, without fail, there is something interesting.
I had a hunch that they might be some sort of prize broker, but it was odd that Scrabble never* had major prizes, only consolation prizes. Perhaps Exposure Unlimited was subcontracted for the fee-plug portion of the show. And maybe some or all of the prize budget was subsidized from Exposure Unlimited's placements, thus the mention.
Interesting that Scrabble mentioned the prize schlepper, when other shows with a great deal of schlepping did not.
The Burbank-based Exposure Unlimited linked above seems to be related to the 1980s one, and perhaps they retooled their business after the prize wrangling market subsided. That said, it is odd that an advertising company does not have a web page, so either they have been acquired or are no more.
(*Whenever I speak in absolutes, someone pulls an exception out of their uh, archive, within 3 posts.)
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EDIT:
And apparently they were acquired/merged with the aforementioned Promotional Consideration, Inc. The address listed in Modor's link above matches the address on PCI's website. (At the very least, they moved into the same building.) Mystery solved!