When a value is selected for B2, auto-populate C2 with date that selection was made

[SnakeDoctor]

I am trying to do something I thought was simple, I need to have the date populate a cell when a selection from a dropdown is made in another cell.



Details:
Column B cells are set as a dropdown list of employees.
Column C cells represent the date the Column B selection was made.
Date cannot change every time the spreadsheet is opened.

Attempts and Failures:
(with Column C cell Format set to Number/Custom: mm/dd/yyyy)
=IF(B2<>"",TODAY(),"") - WORKS but date changes every time sheet is opened

=IF(B2<>"",DATE(),"") - produces "Not trying to type a formula?" error

=IF(B2<>"",DATE(mm/dd/yyyy),"") - produces "Entered too few arguments" error

=IF(B2<>"",DATE(TODAY(),"") - produces 'Entered too few arguments" error

among others too numerous to count, only one works - and it changes every time I
open the sheet.

This is driving me mad, it should not be this hard. It is just a simple 'IF/THEN' scenario! And every search I've done to find the various formula 'date values' produces nothing usable.

PLEASE HELP!