"My put contract says that you have to buy my FooBar shares at $X, despite the last sale price being $Y and $X > $Y." I thought that FooBar would drop down to at least $Y, that's why I bought the options contract.

In other words, as long as I have shares to sell, and there's a contract saying that you will buy them from me for a fixed price, then put options could most certainly exist.

The point is that without short selling option MMs couldn't hedge so no one would sell put options.