The search for Producers is not unlike actors looking for agents. Sure, you can send in unsolicited manuscripts like an actor sends in headshots, and hope that yours falls into the right hands. But the chances of that happening are the same as winning the lottery . . . twice in a row.
The best way to get a script through the door is through a recommendation. Have someone else that the Producer knows (and respects) be your mule and march the script across the border of the Producer's office for you.
Even better than that is getting your show up on its feet. Plays and musicals were meant to be performed, not read. So get them up and get people there, by any means necessary.
The other thing I tell writers is not to worry so much about getting a Producer on board early. If you've got one, great, but if not, don't sweat it. I know it may seem like a Producer is an answer to all of your administrative and financial problems, but that's not the case. That's like saying a babysitter is better taking care of your child than you are.
So, I tell writers the same thing I tell those actors looking for agents. Just keep doing great work and the right people will take notice. When you are ready for an agent or a producer, one (or several) will be banging down your door begging for the chance to work with you..
Isn't it better to have people begging than begging yourself?