What does this mean?

Usually a drought in this case refers to either dating (not finding a suitable partner) or sex (not having any)

She is using the imagery and metaphor of a *drought* to refer to her current situation. You’d need to know the context of her situation in order to know specifically what she is referring to. Usually, you see this in terms of dating/sex/relationships. So it would mean that she finally had a date, finally had sex, or finally entered into a relationship — after an absence or period of time without those.

It is indicating that something that she has wanted for a long time has finally happened or that she has gone a long period of something not happening and it has finally happened. Will need context for exact explanation though.

She was dry down there all the time but finally found the kinks that make her beaver wet

It means she’s finally getting laid after a long time.

she has finally showered.