Ok, I try and keep this short.
I suggest that you attempt to re link the tables, but WHEN you create the links, two things to note:
First up, MAKE SURE you link the tables using a FILE DSN. it turns out if when you from access launch the odbc connection system, it defaults to FILE. So, do NOT create this ODBC link from the control panel, but do this from access.
don't create/use a "user dsn" and do not create a "system" dsn. Make sure you choose FILE for the connection.
This one:
Next up, be 100% sure, in fact 200% sure, when you linked the tables you select this check box:
this one:
If you follow the above, then you can distribute this applcation (often called front end) to each work station and it should just work, just connect, and should not prompt users for a logon.
And you do NOT get that [x] save password when you re-link - ONLY first time you create the table links, so you may well have to re-create the links.
The other issue of course is the ODBC driver you choose. You can choose the built-in long time "legacy" "SQL Server driver", or you can choose a newer and later Native 11 - 17 (or even 18) now. But, if you link using a newer driver, keep in mind that these newer drivers are not installed on all work stations by default, and thus you HAVE to install that same driver on each work station for this to work.