Awful skin problem on hand

I have these dry, cracked skin patches (4 of them) on the palms of my hands.
No matter what I do they won't go away! I've tried every type of lotion and
oil under the sun; changing my diet, drinking more water; wearing rubber
gloves when I shower and wash.

I even tried ripping off all the dry skin, cracked skin, (which hurt like
hell) then covering it with neosporin and a bandaid. I went weeks doing
this, keeping it constantly covered and moisturized. After a while the
wounds healed, but the skin didn't grow back! I can see the surrounding,
healthy skin, then where the patch was is a "shiny patch" of skin (ie, no
fingerprints on it). I decided to stop using the bandaid on one of the
patches, but the skin grew back dry again! While the others have shown no
change!

ANY ideas to fix this would be appreciated. Thanks.