I am using a formula of:

=DATE (YEAR(TODAY()), MONTH(TODAY()), A1)

A number (day of the month) is in Column A and this formula is in
Column B.

Inserting just a number in Column A will show the date in Column B.
i.e. 2 Nov 06

But, doing this in an older Wks will show the date as ONLY 31 Oct 06. I
have tried a bunch.

A new WKS will be correct in the date.