For a start, what usually wets materials is contact with a liquid. A liquid is an enormously concentrated source of wetness. Try to wet anything with vapour, and you will discover just how low the concentration of liquid in air is, in comparison with say, a glass of water.

Secondly, there is the fact that a wet object, such as a soaked cloth or dog, generally gets dried by dripping, shaking, soaking up moisture with a dry cloth, or by evaporation. The last-named is far and away the slowest, even if it is the way in which the last traces of liquid must be removed.  But evaporation works by the same route (only in reverse) as wetting by vapour. To dry a cloth by evaporation in as few seconds as it took to soak it, you would need to heat it explosively. That not only is difficult, but would be likely to destroy the cloth.

Wet dogs are self drying, up to a point, though they are inclined to wet everything in the neighbourhood. As Ogen Nash said:

"...And I have found by actual test

A wet dog is the lovingest."

Translatrix, that was a very sensible question. However, not all is as simple as one might think. For one thing getting rid of all the air is not as easy as it looks. Apart from bubbles caught between fibres and in hollows, some air remains sticking to some bits of the fabric, especially oily bits,the bits that resist wetting by clean water.

You might find it helpful to think of the dry fabric as being "wet with air", just as fabric removed into the air from water is "wet with water". However you certainly are close to right about it being easy to wet the fabric by dipping it into water, possibly with a bit of squeezing and swirling.

Part of the reason is that, depending on the nature of the fabric (whether it is oily, for example) the water, either by itself, or with the help of detergent, attaches itself to the surface of the fabric, or even soaks into the substance of the fibres. Generally it does this more strongly than the air molecules do. Once this has happened only the most superficial part of the attached water can reasonably easily be removed by gravity, squeezing, or spinning. The remainder, that part which makes the fabric feel damp, can only be replaced rapidly by washing it out with competing liquids such as dry alcohol, or by evaporation.

And in that process we are back to the slow options!

Does that description help? If not, please ask again.

Go well,

Jon