I can't think of the single you word you might mean, although "wringing dry" is a fairly common metaphor to mean something is being exploited to the maximum degree possible, for example "Disney is really wringing the Star Wars franchise dry!".
Another common word for the overuse of something in a negative way is "milking" - for example, "Disney is really milking the Star Wars franchise for every last drop!".
To be honest, 'overexploit' is not a word that is often used these days, mainly because the word 'exploit' has come to have an almost entirely negative connotation, so to use 'overexploit' implies that there is an acceptable degree of exploitation. 'Overuse' is fine, as 'use' is not necessarily negative, but it doesn't fully fit your context. I suppose you could say "overmilking", or add the prefix 'over' to just about any suitable verb to suggest it has been done excessively, but the same thought would apply - if you can do something to excess, there must be some reasonable degree. Personally, I wouldn't suggest "overwringing" as it isn't a recognisable word and 'wring' sounds too much like 'ring', so it would just cause confusion. And you couldn't say "overwringing dry" because 'dry' is as far as you can wring something anyway.