The original expression is "to dry up". Your memory of your lines evaporates and leaves you like a dried-up and useless thing, unable to perform. But native English speakers are fond of turning verbs into nouns (for some reason), so "to dry up" turns into "to have a dry". And then we turn the noun back into a verb (for some reason), giving "to dry". More info: https://www.theguardian.com/culture/2001/sep/05/artsfeatures.theatre

I believe that dry in this case refers to the moment when an actor/actress is on the stage and can't remember his/her words. This is called "dryness", or to be "drying on stage". For example, I found an article about this feeling that says: "Yet every actor has experienced the awful sensation of "drying" on stage - that moment when, inexplicably, a line or word refuses to come to hand when you need it".

Speaking from a theatre background, yes we typically say an actor "dries" when they forget their lines. Steve's answer is especially helpful - yes, we can use adjectives as nouns and so on.