Not a bad idea, but even the most extreme ice ages on record didn't so much take water out of the oceans (only dropping sea levels a few hundred feet, not even a significant fraction of 12000 ft on average), as much as cause sea ice and glacier cover, not to mention it not gybing with the MM climate. The worst case scenario is a snow ball earth, but even then there is some evidence it was slushy around the equator and oceans still were underneath.

IMO there might have been a brief nuclear winter maybe 10 years or so (it explains all the dust storms), and then warming predominated again, that seems a likely scenario to me for what happened (sadly all too possible).

I suspect the 'water wars' mentioned were (and will be) over fresh water. With warming glaciers and snow pack, their runoff and many of the lakes and rivers the feed are dwindling and disappearing. If you also consider nuclear fallout (which rains into the hyrdologic cycle), combined with air and ocean currents, it is quite likely virtually all above ground sources of fresh water would get contaminated to some degree.

How much the ocean would dilute it (AFAIK any radiation would be from particles caught in suspension or in microscopic animals), and whether it could be made drinkable is to me a good question. I don't know if fallout would or could contaminate sea water enough to make distilling it too dangerous. If it isn't, life could be preserved and maybe flourish again from the sea, and seasteading (a little like a much shallower water world) might be a good survival strategy. It would certainly make an interesting sort of crossover or future direction for the films settings.

If I recall from Thunderdome we saw the ruins of the Sydney opera house, so I am confident about Max being Australian and the setting, but I don't know if we got a good look, or if Miller has ever mentioned the oceans, or had Max drive by the sea shore. A pet theory of mine is that he hasn't, and that the area were he met the children in Thunderdome was a lot like the area from the film Rogue (the Kakadu Wetlands). Australia has so much to offer for future MM cinematography, it would be awesome if he visited a place like Coober pedy, or Alice springs, or traversed the snowy mountains.