Therefore I conclude that offensive molecules are attaching themselves to the H[sub]2[/sub]O molecules as they become gaseous.
But the water molecules are so small and the organic molecules of offensive odour must be quite larger, particularly if they are organic, and possibly way way larger if they contain DNA.
Can molecules of several orders of magnitude larger than hydrogen dioxide really piggyback on the water molecules?
Or what is science on the transmission of scatological odour?
I don’t have the factual answer but I want to poke one of your assumptions, which is cause and effect between dryness of shit and odor. There may be a correlation, but the cause could be, for example, an expenditure of the gases that cause odor coincident with the expenditure of water into vapor. The odor components are not necessarily attaching itself to water vapor to make their way to your nose. Poop also emits a fair amount of methane.
The smell of shit (and farts) is due to various nitrogen and sulphur containing compounds such as indole, skatole, mercaptans, and hydrogen sulphide. Some of these are gaseous and volatile and diffuse naturally through the air. See The Science of Stink. The molecules are not that large, and certainly don’t contain DNA.
Note that methane itself is odorless. It is other compounds that produce the smell.