1837 representing U.S. colloquial pronunciation of shut.
Online Etymology Dictionary, © 2010 Douglas Harper
Examples from the Web for shet
Lyddy Ann she walked through the entry an' into the fore-room, an' he shet the door.
You jest pull in my sheets, an' shet the winder, while I do it.
Did you ever see anything like the way he got shet o' that drummer?Oh, You Tex!
William Macleod Raine
I says, "Now, folks must shet their eyes—don't open them a crack!"With the Colors
Everard Jack Appleton
Shet your mouf dis minit: ef you don't I'll sell you; 'fore God I will.Clotelle
William Wells Brown