Is water always a product when an alcohol is oxidized

It will depend on what you mean by "oxidized" and on how the reaction is carried out.

If you are oxidizing the alcohol to CO2, then, yes, H2O is always a product. When you are oxidizing --OH to =O, most likely water will not be formed. But here, too, it will actually depend on what the oxidizing agent is.

Oxidation of a primary alcohol results in an Aldahyde, 2 molecules of primary alcohol oxidized results in an ether, oxidization of a secondary alcohol end product is a ketone.

No oxidation just means you turn the alcohol into some kind of carbonyl by removing the hydrogen. There are many ways to do this without producing water as a product