Set Variable [ $Name ; Substitute ( YourTable::NameField ; "\"" ; "\\\"" ) ]
Thank you for your post.
I am unable to replicate the issue. Here are the steps I took with FileMaker Pro 13.0v3 under Windows 7:
1. I created a file "Test.fmp12" with a Text field named "fullname"
2. I entered several records and two records contains: James "Jim" Doe
3. I created a script with the following information:
Set Variable [ $name ; Value: Test::fullname ]
Enter Find Mode 
Set Field [ Test::fullname ; $name ]
Perform Find 
4. I make one of the James "Jim" Doe records active and execute the script. The two records are found.
Let me know what I'm doing differently than you so I can replicate the issue.
I suspect that Sam Jaques may be using a stored find request that refers to the $Name variable instead of using the enter find mode/set field method.