Elementary teacher
I don't know how it works there, but most schools are governed by a larger school district. You will have to find a school district williing to go through the process with you. I filled out all of the paperwork myself, had the superintendent read and sign it. I paid the fees myself (public schools are exempt from the huge I-129 fee) by purchasing a money order and having the Superintendent sign it. In my experience, you have to "sell" the idea to your superintendent and make it easy for him/her.
Perhaps, with Premium Processing, you could accept a full-time job and have everything get approved in time for the start of the school year. That wasn't so in my case, so I worked as a sub first.
I don't know of any placement agencies. They may exist - I don't know.
I would start as a sub. You can always easily transfer to a different school district if an appropriate opening comes up later, too.