From my past experience, dishwashing/presoak chemicals have a LOT of nasty things in them. These alkali/acidic chemicals are not friendly on the hands. From this point on, you need to wear gloves, you really don't want to damage your hands any more than you have done so far.
Look for elbow-length gloves for now, and make sure that you put a thin coat of lotion on your hands before you put the gloves on.
Best of luck with that...
-SF
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