I used a BATFE 5320.20 moving from AZ to WA, Came back signed approved after a few weeks. Not sure it was necessary but I figured it didn't hurt either
It would be wise to first establish that your trust will be valid in Oklahoma and then yes, a form 5320.20 to update the registry. I am NOT an NFA attorney, hope you like OK.
Look at the bottom back side of your stamps. Instructions printed right there. I can't believe you didn't see that (me being smart ass). No form needed for permanent move, form 5320 is for temporary transport.