The same exact word is found in Job 38:31, and over there it is clearly rendered as "bundles" "chains" or "shackles"--"Can you bind the chains[b] of the Pleiades?". This is the basis for the traditional Jewish translations.
The academic translations probably take the root of this word to be "עדן" which refers to fattening. The word מעדנים is frequently used in Tannaic and Mishnaic hebrew and refers to delicacies, since they fatten the belly. see also Proverbs 29:17 where this word is found. So in the context of Agag walking מעדנת it would mean something like "he walked unsteadily like a person that just finished eating a big meal", hence the "faltered steps". The source of this translation is Targum Jonathan.
It is interesting to note that some (David Kimchi) interpret the Targum Jonathan differently, according to them Agag walked "calmly" as if someone finished his meal and is satisfied; this is the opposite of walking unsteadily. According to this understanding the verse is saying that even though he was walking to his death sentence he was walking calmly and happily "like the manner of kings".