Those are some good suggestions. I found it pretty strange when some recruiters asked me for scans of my passport and documents.
They don't believe me? Is that something I can really lie about? At some point, if I'm not really American, the Korean Embassy in D.C. may provide comment on the subject... But I digress.
I don't have a scanner nor does anyone I know. (I live in the sticks, y'all).
BUT could I take a digital photo with my camera? I reckon if I'm handy with it they'll be none the wiser, eh? Getting directly over the document may take some ingenuity, but I've got some time on my hands.

You can take a digital photo, but if your local library doesn't have a scanner, I'd be stunned.
And on the documentation/proving your nationality front, if you think this is over-the-top, you are hilariously unprepared for the hijinks that await you.
Also, it sounds like you're looking and applying for jobs on your own, which is fine, but if you're having trouble, why not talk to a recruiter? Like a real one. A recruiter that only forwards you an SMOE application isn't a real recruiter-- it's someone with access to google.
I used Aclipse Recruiting, a friend used English Apple and got to Incheon, there's tons of recruiters all over the place (Footprints, someone's always posting in the job section for Tiger English, these are just pulled off the top of my head, never mind what treasures google could bring you). I got turned down from a hagwon with a recruiter (unprofessional picture, kiss of death, even for an American white lady like me), but then got a PS gig through the same people. Work with someone for a while and see what happens.
Also, if you're interested in applying for public school earlier, rather than later, is a good thing. Find out when the open date is (again, google, start a thread here, or ask a recruiter) and literally get your stuff there on the first day.