Difference in terms- Class III vesus 03. Class III refers to National Firearms Act weapons, such as full autos, short barreled rifles, etc. 03 is an 03 C&R FFL. If you have the latter, notify the licensing center of the ATF of you change of address. You can MAIL rifles and shotguns to yourself at your new address. HANDGUNS may not be mailed except by an 01 Dealer or a Manufacturer. Us C&R FFL folks cannot legally MAIL a handgun. We CAN ship by UPS or FedEx- they require next day air. You will be better off finding a dealer that you can pay to MAIL your handguns for you.
Full autos- contact the BATF- you are required to notify them when moving those across a state line.