besumei

It's well known that there is an obligation to get drunk on Purim. The source is the gemara in Megila (7b):

מיחייב איניש לבסומי בפוריא עד דלא ידע בין ארור המן לברוך מרדכי

Interestingly, the verb for intoxication isn't from the root שכר, but rather l'besumei לבסומי. This word comes from the same root as perfume or fragrance - בסם or בשם - bosem. This is an ancient word in Hebrew, and appears in the description of the incense brought in the Temple. It is also familiar to us from the besamim we smell at Havdala, after Shabbat.

The root bosem made its way into English as well. The balsam tree got its name from the Greek balsamum, which derives from the Hebrew basam. Later, balsam led to the word balm, and balmy - fragrant or mild weather.

What about the other meaning of balmy - insane or foolish? There seem to be a few theories. One theory claims that it is a corruption of the name of a lunatic asylum by the name Barming Asylum in Kent, Britain. A different approach is that it comes from the word "barm"- the frothy, foamy head found on a glass of beer or ale, which derives from the Old English word "beorma." "Barmy" first appeared in the 16th century in a literal sense meaning "foaming," and by 1602 was being used to describe someone acting in an excited or irrational way whose head seemed to be filled with froth.

So while you might be feeling balmy while you are m'vusam - there might be a connection, but it's not etymological.