Maybe, but good old Uncle Bill would probably skip most of the sweet talk and go straight for the drugs.
I don't think that's what is implied in the song. If it is, I find it hard to believe that no one picked up on it for 70 years and after about two-score cover versions. There's even a Lawrence Welk version of this: they kept the part about her agreeing to half a drink more, but took out her having another cigarette. Besides, I think the woman wants to stay --she had had a good time and his persuasion was also working-- and her objections to the man are also her thinking out loud about the reasons she shouldn't stay. "What's in this drink?" was apparently a common expression in the 40s used when people wanted to blame their actions on having had too strong a drink or too much of it.
I like the song, especially Dean Martin's version, because I find the style interesting. I could get through Christmas without it, though, because it is passes as a Christmas song only because the popular observation of Christmas has become more and more centered on winter party time and less and less centered on --well--Christ.
Agree: about Christ Mass, but also about Dean Martin.