Why didn't you get the original refund?  If the IRS or state kept it for something you owed you still include it as refund you received on the amended return.  You still got it it just went to something else.  It was for your benefit.

There was an error on a state form (IRS is not involved). The state found the error and will not issue a refund until a corrected form is submitted.

While the amended return ( form 1040 ) is like a new return with the updated facts, the form 1040-X is a reconciliation  i.e. it assumes that the original return has been processed and refunds issued  or dues have been paid.  This how IRS sees it.

Given the above I do not understand  your statement 

"overpayment from original tax form which has not and will not be paid"

IRS is not involved. This is a state form. The state found the error and will not issue a refund until a corrected form is submitted. (I also believe this is a Turbo Tax internal error dealing with populating forms from earlier forms, the incorrect number was not one I entered--I found one of these types of mistakes a few years ago.)

So do they want you to send a new original return or to actually amend?  What state?  Exactly what error?

I don't know what the legal definitions and nuances are. The term used over the phone when I talked to the New York tax office was "amend your return and submit it." Form NYC-208 had an incorrect value on line 20 which was prepopulated from form IT-196 line 6 (which was correct).

revise your state return to be correct and ignore the X-form.
submit  the corrected pages of the return, according to the instructions you got. it sounds like just one page is incorrect.
make sure you are mailing  back to the address specified in your rejection letter.