How to make my ice blend more slushy-like?

You are fighting against entropy here because the heat of your ingredients and the heat generated by the friction of the blender’s blade are spreading out into the ice, melting it and giving you a homogeneous liquid.

So you have two things you want to reduce here: the total heat and the heat transfer to the ice.

Reducing the heat transfer to the ice is hard with shaved ice because it has an incredibly large surface area: hotter liquid can get everywhere in between the little tiny shavings. This causes rapid equalization and I do not see how you could solve that other than adding ice cubes.

For the total heat problem, try getting everything as cold as possible. This means putting everything into the fridge/freezer (even the blender’s jar if necessary) beforehand and minimizing the time it spends on the counter. You should also try blending for only the bare minimum. If everything still melts too fast, you could try freezing the milk or condensed milk (in something which will not explode when the milk expands, providing extra ice.

Colder ice.

RedBorgs solution of putting everything in the freezer is reasonable, and you already have a freezer. That should get everything to about -20C.

An alternative is to make your ice colder using dry ice. Regular ice from the freezer is -20 C. Dry ice is -78C. Purchase some dry ice and then put your shaved ice in with because I am not sure how the shave ice machine will do with -70C cubes. Or maybe it will do fine and the cubes will be easier to take back of of the dry ice. The water ice will get as cold as the dry ice. Probably cold enough to also freeze the other ingredients when you add them.