Click here and a link will take you to a page with some 200 words/phrases - including your own - meaning 'drunk'. If you study the list, you'll see that hardly any of them is susceptible to rational explanation. Nor is it very likely that an actual source will ever be isolated.
'As drunk as a rat' is centuries old, but why/where/when that became the modern 'rat-ar$ed' I don't expect we'll ever know!