Nice job! The only catch now is that you need to inspect the packages and do a condition report, or if you are lucky, the gallery staff will help you. This is the gold standard. Who knew the government had so many resources for this?
The linked document is probably about a thousand percent too in-depth for your purposes. Generally, take photos of the artwork at each stage, such as unpacking, inspecting for damage, and taking pictures of anything that seems odd. There is a high likelihood that even if your work were damaged, you'd never be able to get any money out of the shipper, but it is always best to cover your ass when it comes to things like this.
Once you have your mountain of photos, get them off your phone. Chances are you are somewhere fun, or at least different right now if you are doing this, and losing your only viable option at a claim with the shipper because you were out drinking and messed up your phone or accidentally deleted them to make room for photos of your dog is a pretty lame way to lose out. An offsite server or fancy tech solution is best, but who has the time or know how to set that up? Just email them to yourself for the time being, or shove them in your Google Drive. You never clean that out anyway, right? They'll likely last as long as your interest in the piece is questioned. So you should be fine