Palmolive dish liquid had a ubiquitous commercial in which a dowdy manicurist name Madge would tell her clients, "You're soaking in it," about the dish soap. That is really my only association with the name Madge, other than it being a flippant name they call Madonna in the tabloids. Given that it rhymes with vag, I'd avoid it as a nickname. Margaret has a lot of cute nicknames, so I can't imagine choosing this one. Even Marge is better than Madge.
I really dislike Madge and Marge as nicknames for Margaret - they are horrid and dowdy and unworthy of beautiful Margaret - Madge was an old vulgar name for a certain part of the female anatomy, certainly used up to and during the eighteenth century in Britain. Madge seems reminiscent of 1920s and 1930s boarding school jolly hockey sticks characters in books for girls. Marge reminds me of margarine.
Still trying to get used to how it sounds, though. I do like the spelling.