Okay.
(a) I agree with insomniak's evaluation -- the whole idea of Joe being lost, but on the way to becoming a new person, and discovering that new person. The man is left without the fibers of his life as he knew it. He has to go on and find what's right for him.
(b) There isn't any concrete indication that he goes to Washington. He disappears -- drops out of the plotline. We don't KNOW where he does. Just as you say there's nothing suggesting him wanting to find himself, I pose to you that there's nothing directly suggesting that he goes specifically to Washington. He just goes...
away. Maybe this is similar to what Kringas and I had said about not knowing specifically whether or not Louis has AIDS because ultimately for this small piece of what we get of their lives, it doesn't matter all that much. Perhaps the specifics of what happen to Joe aren't imperative, so they aren't there. Maybe we're meant to assume certain things, but only know that of the grand scheme. I think to say that we can safely assume he went to Washington simply because there's nothing to indicate that he didn't, or to indicate where else he might have gone is a bit narrow. It sort of kills the notion of a play being very open to interpretations -- works like AIA probably aren't so "well, if he didn't do option a, then he HAD to do option b." They're not so this-or-that.
Being that the play's message is one of hope, I'd like to be able to conclude that Joe does go off to do some self-exploring. Why it is perfectly possible that Kushner left him out to sort of "shun" his character in a way, to me, it jives better with the rest of the play for him to be lost to exploration. Kushner, as I see it, has sort of turned him lose in freedom to go find who he is supposed to be.
I've always found Harper to be the character possessing the most intriguing insight in the play. She says: "Nothing's lost forever. In this world, there is a kind of painful progress. Longing for what we've left behind, and dreaming ahead." She has hope. And I think that's what the play displays overall, hence why I think what I do about Joe's outcome.
A work of art is an invitation to love.