So my opponent has Drowned Disciple in play no cards in shadows and I have the new Ser Mandon Moore with two cards in shadows.

I play Valar does his Drowned Disciple still go on the bottom of his deck and I mill three cards? Or does Drowned Disciple go into the dead pile?

All saves are "Response" effects that use the word "save." More over, all saves interrupt the initiation and the resolution of a kill effect so that the saved character doesn't die and stays in play. Not only does the (constant) effect of Drowned Disciple not use the word "save," it dies and leaves play, so it's hard to understand how it could be seen as a save.

The effect of Drowned Disciple is a constant, replacement effect that doesn't stop it from being killed, but rather redefines what you do with the card when it is killed. It is not a save, since the character does, indeed, die. It will go to the bottom of the deck when it is killed, and opponents will discard.

Oh, so then how does one deal with the cards like that?

Discard it instead of killing it. Blank it first before killing it. Wipe out their resources so that they don't really want to pay the 3 gold to play the character in the first place. Build a recursion deck so that, on the off chance that one of the 3 discarded cards is so vital to your strategy you can't live without it, you can get the cards back from the discard pile. Suck it up and "play through" the lost 3 cards. Things like that.

Not all cards have easy counters. Sometimes its about mitigating the damage instead of figuring out a way to prevent it from happening.