I speak American English.
Typically, we call a tumble dryer/spin dryer simply a "dryer". For the verb form, you could just use "to dry" or "to tumble dry."
I have clothes in the dryer. = I have clothes drying right now in my tumble dryer/spin dryer/electric dryer.
I need to dry some clothes. = I need to move wet clothes from the washing machine into the dryer.
So if you're speaking to an American, you can assume that we know what a "dryer" is without being any more specific.