"I wanna get wet in rain with you" Is there a better word than the vulgar-sounding "wet"?

Maybe "soaked," but I think that "drenched" is actually better.

I wanna get rained on with you

Maybe I want to just stand in the rain with you. The just emphasizes the value you place in the experience of being with the person you describe, no other motives intruding.

You could substitute other terms for stand, such as soak, bask, wallow, play, or any other non-sexual term.

The only reason to say it like that is to refer to the innuendo. If you do anything in the rain, you will get wet so mentioning it is pointless unless it is an innuendo. That said, you could go with soaked as drenched feels negative. The word "soaked" isn't as likely to make someone's mind turn to the gutter. Though however you word it, the audience is going to make a difference. If they tend to fall into the gutter to begin with, there is no avoiding the reference with such a wording. However, if they are more... innocent minded, they may not even notice the reference with the obvious "wet".

You could also change it slightly. Maybe to run, walk, laugh, relax, sing, play (ok maybe not that one), etc... Then leave the rain there and it will imply that you are getting wet without having to say it. The idea is what picture it brings to your head. Simply mentioning rain, for most people, is enough to bring the wet picture you want.