Hello. Thank you for contacting the Toolset support.
Well - default search and Toolset text search is bit different. It will not update the default "s" URL param directly.
Can you please send me admin access details and let me check what workaround I can offer.
*** Please make a FULL BACKUP of your database and website.***
I would also eventually need to request temporary access (WP-Admin) to your site. Preferably to a test site where the problem has been replicated if possible in order to be of better help and check if some configurations might need to be changed.
I have set the next reply to private which means only you and I have access to it.