We moved recently from NY metro to Austin Texas. Our stuff went into climate controlled storage, and the higher value items into a vault, until we pulled the trigger. There were some items we needed for living as soon as we arrived in Austin so we shipped those ahead via UPS. The cost was pretty astronomical, but part of that may have been the value placed on the contents.
I personally packed a little over 5000 LPs for moving. I used double walled LP boxes from Bags Unlimited, but assume similar boxes are available from other suppliers. The boxes were packed pretty full, with some additional filler on either face end, then taped by the mover.
Once we bought a house in Austin, we gave the movers the green light- all the records arrived fine.
If you aren't using a professional mover to do your household stuff, your approach may have to be different, but aside from the weight (movers charge by weight), there was little to worry about. They also handled all my hi-fi gear (properly crated or boxed) without any apparent issues.
Congrats on your move.