During this time also try to interest publishers by letting them know about the play, try to get them interested in following the growth of the play through its different productions. Don't send a publisher your first draft. Even if they ask for it. Most playwrights writing for young audiences do not have agents. There's just not enough money in it to interest agents.
Some playwrights hire entertainment lawyers to negotiate individual contracts.
Those who belong to the Dramatist Guild can have a lawyer look over contracts for them. A good publisher becomes your play's agent, negotiating and collecting royalty fees for you. They split book fees and royalties with you. They do NOT expect any advance payment FROM you. It has a lot to do with whether or not that person feels he or she can sell this play to the public.