A (( HUGE)) thank you to everyone who's commented with tips on this post - my son's okinuma headset adapter bent after his younger brother dropped a device with them still plugged in, and our oldest was more worried about his younger brother not feeling bad than his own loss of the prized Christmas gift - so, needless to say, mama was on a mission to fix these headphones!
This thread had so much helpful info, I just thought I'd throw in a couple of the things I picked up along the way for any complete beginners out there - who, like me, are coming into this with like, ZERO soldering experience!
Because these headphones have a mic, as others have pointed out above, you need to make sure your purchase the proper replacement adapter - you'll be looking for a TRRS, or 4 Pole adapter.
Personally, after I'd watched several vids showing how to solder the existing headphone cable to the replacement adapter, I chose instead to purchase an inexpensive 4 pole headphone cable extension (one end male, one end female) and to simply cut off the unnecessary female portion and wire the extension wire to the existing headset wire (so as to give me wires to work with rather than such a small surface to solder on AND to lengthen our headset reach!).
The Okinuma headset wiring schematic wasn't posted anywhere that I could find online - but the pointers in here helped me to narrow down the possibilities!
Once the broken adapter and sheathing were removed, wires were:
+Blue
+Green/Black Stripe
+Red/Copper Stripe
White sheathing with +Copper wire w/ a Red wire running through the center of it
Meanwhile, the extension adapter I'd purchased had the traditional:
+Blue +Green +Copper + Red
I had to guess and check to see what each of the headphone wires were - after stripping away the plastic outer sheath on each, I simply twisted together the wires I believed corresponded with each signal.
* This will not give you any sort of result of trying to test like this! **
After growing frustrated with the lack of ANY results after like 5 attempts, I found this thread and found another person who was having problems testing this way, but didn't see anyone clarifying that **THIS WILL NOT WORK** (I believe) this is because each wire is coated in a sort of enamel to prevent the wires from shorting while encased alongside the others.
So when the YouTube videos show someone heating the wires with a lighter or just the soldering iron itself before actually soldering, I believe it was to remove the enamel and also to burn off the thin fabric fibers that are interspersed within each super thin wire, to prep the surface to receive solder.
Another wonderful thing I learned in this thread was that the copper wire in the separate white, microphone sheath is a ground and will need to be wired into the main ground - so total, I had three wires here to join together (ground from adapter, from microphone sheath, and main headphone ground)
Lastly, after all that was figured out, Alexandria above gave me the last bit if advice that saved the day - the red wire from the white microphone sheath is covered in a SUPER thin, transparent plastic that will also need to be removed before soldering!
After many trial and errors, I certainly learned a lot, these were the connections that worked for this particular headset, in case anyone needs to know:
* Blue to Blue (Left)
* Green to Green/Black Stripe (Right)
* Red to Red (Microphone)
* Copper/Red Stripe to copper from microphone sheath to copper in adapter cable (Ground)
* I hope maybe my beginners insights might serve to help someone else super new to this! Thanks so much for all of the help!