No need for you to try and contact the State of Maine. Here's what you need to do. Contact the University of North Carolina at Chapel Hill. Ask them for a letter stating that "you completed an approved secondary math program for teaching high school math". This letter must be on official letterhead stationary and signed by the Dean or whoever oversees their advanced math programs.
I believe that once this is done, the worst that could happen is that you would need to take the "Exceptionality course" and the the Praxis II. Thanks.
Oh my goodness, what a huge burden lifted!
I still need to track down the Exceptionality course they say I need but which I took (and which, to be fair, is not apparent from my transcripts), and show the State that my Praxis scores are so old they've been purged by ETS. But all the work I was already putting in motion -- getting my undergrad transcript, requesting course descriptions for the engineering courses that contained math, etc. etc., has been put aside for now.
I knew rational me was right. I just couldn't hear her through the panic.