Did you actually remove the sticker covering the middle two pads?
As @Ganblejs said, it's not just skin irritation (which is rare) but corrosion of the contacts, which is what you have there. With the middle two pins covered you don't get the corrosion.
Washing the back of the watch with alcohol would actually remove oils from your skin which may have otherwise helped to protect the contacts, so that won't have helped at all (but it won't have actively dissolved the contacts).
Once the outer two contacts are gone, that's it - you won't be able to charge the watch any more. However you do still have half the contact left - all I can really suggest is to immediately cover those middle 2 pins, and that should stop it degrading further.
It's possible you may be able to find some conductive paint you can paint over the contacts with to re-cover them and allow you to charge reliably again?
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Did you actually remove the sticker covering the middle two pads?
As @Ganblejs said, it's not just skin irritation (which is rare) but corrosion of the contacts, which is what you have there. With the middle two pins covered you don't get the corrosion.
Washing the back of the watch with alcohol would actually remove oils from your skin which may have otherwise helped to protect the contacts, so that won't have helped at all (but it won't have actively dissolved the contacts).
Once the outer two contacts are gone, that's it - you won't be able to charge the watch any more. However you do still have half the contact left - all I can really suggest is to immediately cover those middle 2 pins, and that should stop it degrading further.
It's possible you may be able to find some conductive paint you can paint over the contacts with to re-cover them and allow you to charge reliably again?