As I said before, I came to Korea at the end of March for a teaching position that didn't pan out after a few weeks. It was initially considered a mutual parting of ways, with LOR given and payment for work promised on the next appointed pay day, yet quickly became a much more complicated situation of dubious allegations, unnecessary Kakaotalk harassment,and threats of lawsuit over supposedly lost revenue. The initial complaints were very weak such as a light accidentally left on in my company apartment and bedding missing, but that was just the tip of the iceberg. It was soon alleged that I costed the director and more importantly the owner a lot of money in returned tuition fees from students leaving because of me (mind you when I left the position both my morning homeroom class and afternoon hagwon class were filled to capacity, having grown rapidly since my arrival).

The director who was the one very rudely contacting me on the weekend outside of normal business hours with the allegations and had just handed me my LOR on good terms two days prior said she needed to meet with her boss the owner who was supposedly feeling completely disrespected (he doesn't speak English and couldn't understand me at all when I spoke English during meetings, so I have no way to verify how he felt about anything and not stupid enough anymore to think the director was correctly relaying what I said), would follow his decisions, and she'd let me know how that went.  The day I arrived for my current position, the director contacted the former employer to request copies of my visa documents ( which the director said they no longer had) so the person mentioned above knows where I currently work. I was also asked if they'd paid me and said I would be paid on May 4th, but that didn't happen and despite my attempts to contact the director of my former school have received no replies. The director of my new teaching position though has again asked me what wages I was paid and documentation for tax purposes.

I've already very good advice to never admit anything, apologize for anything, and never sign anything and it doesn't matter if the whole school left b/c of you, you absolutely did NOT do anything worth apologizing or accepting responsibility for. So I ended all contact with my old employer except for Skype being made aware that the company was actively trying to screw me over and I needed to stop communicating immediately, don't say another word and whatever they accuse you of you don't know a SINGLE THING about.

However the fact remains I worked 3 actual weeks for this company and was kept on payroll until the end of the pay period May 4th and feel like my labor was not free, and I only want the director to keep her word and pay me.  Yet I know they will not simply pay me and forget I exist. They don't feel the same way about the situation as it was initially presented to me at my departure.The director has chosen to blame me for the situation when she was undoubtedly questioned about how her decision to hire me didn't work out, and I fear that once contacted by the labor board my old employer will attempt to absolve themselves of responsibility by causing trouble for me now going forward in my current position and making it seem like I was the problem. I don't want to be vindictive or malicious I just want them to pay me then kindly forget I ever existed so I can move on once and for all, but that's too much like right it seems.

Well, I mean, of course they can.
On the other hand, one can sue for libel and slander. On the other other hand, as a foreigner, it'll be really difficult to pursue a libel case, and will probably cost you more money in legal fees and stress than you would gain even if you were to win the case.

As for dealing with your old school, I've heard a lot of similar stories over the years, and I'll repeat here what I've heard others comment:
-I would file a complaint with the labour board: you worked for the school for 3 weeks and was not paid for your time. That's probably actionable.
-It might also be a good idea to have that complaint on record so that if/when your ex-employer starts filing stuff, there will already be prior claims against them. Sometimes the best defense really is a good offense.
-If you aren't already doing so, make sure that you have records of *all* communications between you and the other party. If possible, limit all communication to text form, as it holds up better in legal disputes.

I hope that everything gets settled quickly and in your favour!

It's illegal for them not to pay you. There's no question about that. You can use this service to file a complaint against them for non-payment of wages:
https://www.epeople.go.kr/petition/htp/pff.npaid