How can I create a drinkable ice slurry?
When you mix it, you're bringing everything to the same temperature, and if the amount of heat in your mixture isn't enough to bring all the ice up to 0C before all the water freezes, you're going to get a block of ice.

You could be careful about it, and figure out the temperature of your ice and your water, and get the right amounts, and perhaps even use warmer water (or warmer ice). But you can also just use less ice per water, and that's easy. Simply start with water and add ice, blending as you go, until you have the amount of ice and water that you want, stopping before you add so much that you just get solid ice. Once you've done it once, you'll have an idea how much ice and liquid to start with.

I don't really see why you'd bother lowering the freezing point. 0C is plenty cold for human consumption, and you can just as easily just use more water and less ice. If you do add things for flavor like alcohol, sugar, or anything with fat (e.g. dairy) then you will end up lowering the freezing point, and possibly making it softer when it's almost frozen, but this doesn't really affect the fact that you want to get it to the ideal temperature by controlling the temperature as you create it.

(Note that a smoothie is basically an ice slurry, with a lot of other stuff in it, and they're made just like this - blended with an amount of ice that can't freeze the whole mixture.)