Here is some feedback on what I do and have had no damage and had very positive complements on the shipping.
Break the sxs down to 3 pieces. I use microfiber towels sold at Costco. Wrapping the barrels, forend and rear half in the towels seperatly and securing them with rubber bands.
Leave the egg crate foam in the impact case and place the pieces inside making sure they dont touch.
Latch the case and use a few tie wraps through the lock holes, or use a few pieces of tape over the latches.
Place in cardboard box using scrap cardboard to shim the case in the box so it does not move also adds another layer of protection.
I ship by Priority mail asking for a signature.
Ship on Monday-Wednesday usually puts the shotgun to the buyer no later than Saturday.
I have a USPS account so I ask for notifications and also mark to update buyer by email.
USPS in my experience has been great. Not looking to debate, as others may not have had the problem free experience I have had.
Good luck.