Option One: search out and use a Safe Moving firm. They will have trucks with hydraulic lifts and folks experienced with heavy-duty dollies and the like for moving a safe to and from without causing a lot of damage to the safe, or either domicile. They generally will not move firearms. They are likely to be expensive, especially out-of-state.
Option Two: Sorrt-of DIY.
Contact the people at PODS, and have them drop off a container at your current house. PODS come in standards sizes, 8x10x10. 8x10x12 and 8x10x16. Construction uses a floor of 4x4 with 4x4 for support columns and a skin of metal-clad fiberglass.
REnt an appliance dolly and get some strong friends and get your safe (empty) to the PODS. Than lay it down flat on its back. Use ratchet straps to spider-web the thing to the internal posts in the PODS. Not a horrible idea to put a floor rug down before putting in the safe (to keep it off the plywood floor).
Note, the rig used to load the container on the truck may tip the container to 15º in one of three axes--strap accordingly.
With the remaining volume, you can fill that with all the stuff you'd rather not leave around moving guys. Or books, framed art, etc. (Reloading benches & stuff, too). Now, the secutiry on one of these is a thing metal rolling door and a padlock you provide--it's mostly out-of-sight-out-of-mind.
PODS ships the container to just about anywhere. They bill you by the month for the container, and then, per trip.
Now, I have experience in this, I got most of a 1400sf house into a 16 container:
I used almost every inch inside (and 8 pairs of ratchet straps and two cargo nets)
Hard to see here, but the tube tv is sitting on the end of the safe. Wedged in pretty tight, and no evidence of it having moved at all, even having beeing moved three separate times. (Lots of dunnage in there, too.)
This is the delivery trolley that gets the can off the local-haul truck (and the source of most of the tipping).
My 2¢, Your mileage may vary.