There are many products designed to slow down dogs eating so I googled for dogs drinking and it turns out many dogs have the same issue as yours and there are products to slow them down! There are several different styles: Slow-Down Pet Water Bowl, Dogit Slow Anti Gulping Bowl, I feel like something like this would work too.
EDIT: Just realized I was on /r/Dogtraining and not r/dogs and you were asking for reshaping behaviors not other solutions. Sorry I need sleep! I'll leave this up in case anyone is searching for the same thing.