The Overwrite cookie
option overwrites only the affiliate info stored in PAPVisitorId
in the database of Post Affiliate Pro.
So the value (32 or 40-character long string) of PAPVisitorId cookie in your browser and database of Post Affiliate Pro remains the same, but the affiliate info stored by it is changed if the particular customer clicks another affiliate link and a click is tracked.
If there is a cron job running regarding your Post Affiliate Pro (all hosted accounts of Post Affiliate Pro (network) ) have a cron job set up), then the tracked clicks are processed offline after the cron job has been executed at least 2 times.
When you are testing the "Overwrite cookie" functionality, then you should use the following page to see if the affiliate info in the stored PAPVisitorId has changed or not:
There you will see the userid of affiliate associated with the PAPVisitorId immediately, because in its source code (check it with your browser) there is a block of code including the writeAffiliateToCustomField($field_id) function that forces the particular PAPVisitorId to be processed immediately.