Just wanted to add. My Access on my home desktop has the "good" version of Linked Table Manager (top), yet most of our clients, including this one, have a seemingly more primitive version (bottom). I have no idea why or if it's possible to switch. Even on fresh installations of Office 365 I see the bad version.

You can use VBA to automatically look through all the table's DSN strings and change them.

You can see the DSN by hovering your mouse over a table's name (see picture below).  This string is stored in the Connect property of each TableDef.  Sample code is below:

Dim db As Database, n As Integer, i As Integer, s As String
Set db = CurrentDB
n = db.TableDefs.Count     ' n = number of tables minus 1
For i = 0 to n
     With db.TableDefs(i)
          If .Connect = MyCurrentDSN Then
               .Connect = MyNewDSN
               .RefreshLink
          End If
     End With
Next i

The If statement inside the For loop makes sure the table's current DSN is what you expect it to be before changing it.  After the change, the table link needs to be refreshed (RefreshLink).

This is not a frustrating problem.  Having to switch servers and DSNs is understandable and pretty routine stuff.  And the VBA code above is not terribly difficult.  It would not take 4 hours to run this code even if there were 10 thousand tables.  4 minutes maybe.

But make sure you backup your existing database and make sure you are proficient with the Access topics involved in the above code before using it, I recommend you.  The code is simple but the task is crucial, so you need tip-top knowledge for it.  You would likely need to make big or slight modifications to the code to suit your specific needs.

It worked, thanks! I tried something like that before, the only difference between old and new is that old didn't save the password (to save time - it's 2,000 tables). I found that the password still wasn't saved. But yours worked (I think :))

Correction: the For statement above should be:

For i = 0 To n-1

The password is always saved, otherwise the user is prompted for the password every time a table is opened, which is not ideal.  In my picture above, Access hides the password in the DSN for security (the password is stored right after "UID=sa").