Word-of-mouth leads in Honolulu - first time posting

Two small Honolulu employers told me they are hiring but have not posted yet. I have been trying to stay organised about it. They prefer people who can start within a couple of weeks. What do you write in a cold email?

Thanks for writing this out. I went through something close to it 5 months ago, and it got easier once I kept a simple log of every call and form.