Steven, Sorry for the delayed response.
Both Woodrich-Brand Timber Oil and Wood-Tux can be applied to damp or dry wood. There is really no "minimum" moisture content required. Generally speaking Timber Oil will dive deeper into really dry wood.
When it comes to "maximum" moisture content, Wood-Tux can be applied to wood even if it is completely saturated. When it comes to fresh cut wood that is considered "wet" such as new cedar, Timber Oil seems to do better on the initial application.
HOWEVER
We do not recommend trying to stain immediately after washing. It is always going to be better if you wait for the wood to dry, we are not trying to promote a "deck in a day" type solution. The idea was to create products that allow you to stay on schedule during the rainy season.
When you stain wood that is really wet, it is going to take a lot longer to dry and/or cure.