Alright, I did some more sleuthing. I listened to just about every version of "At Last" on Apple Music, paying attention to the line "a thrill to press my cheek to" at around 1:45. Nearly every one had the problem, including a 1993 release from UMG, which I think had merged with MCA, which if I'm not mistaken, had acquired the Chess catalog. I may be wrong on that stuff though. But it looks like it's on every release since at least 1993. The version on the soundtrack for My Best Friend's Girl actually sounded decent, but the problem was still there, just a little less pronounced. The same with the version on Heart & Soul: A Retrospective, also released by UMG. It was not present on a 1995 K-Tel version, but that was one of the worst sounding things I've ever heard otherwise. There were a couple of versions where it appears as though someone tried to fix the problem by basically removing the "p" sound altogether and hoping the listener doesn't notice.
Nearly every version I listened to was post 1999 though. And every one had the problem, including just random stuff I found on Youtube. Without an original to listen to, though, I have no way of knowing if it's been there since the music was originally recorded. Doesn't necessarily bode well for the upcoming release though. Fingers crossed.