@justej I am sure moving companies have seen it all. I wouldn't worry about it. I would suspect the team that moves you will just go about their work without making any comments about what they see or don't see.
Unlike my friends once who helped me move out of my apartment when I was like 24. I forgot about the bedwetting stains on the underside of my mattress. A few years previously I had an accident without my plastic cover on the mattress and even after trying to clean it the wet stain was still visible. So I flipped my mattress and forgot all about it... until. Before they came to help I took off my plastic cover and packed it away in a box with my GoodNites (I was skinny back then) and taped the box and set it with other boxes randomly. Later as my friends got the mattress off the bed they put it on its side to take it out the door and that is when they saw this very obvious bedwetting stain smack dab in the middle and a little left of center. LOL I was out in the living room area and had not seen it yet. When they came by the clean side was facing me and I noticed they were laughing. I asked, what's funny. One said, didn't know you were a bedwetter? I prob went bright red thinking, how do they know (I was wetting my bed about once a month then). Did I miss a pack of Goodnites in the closet, what went wrong,how'd they find out? Then they turned the mattress over and I saw the bedwetting stain that I had forgot about. I was like Oh that, I spilled a drink in my bed once. They of course said sure you did and laughed a little more. Then one friend said don't worry, everyone has that one stain on their bed from the time they peed the bed as a kid. We all laughed, them more than me who was still sweeting thinking what if they find out.
So yeah, just don't ask my friends to help move you and all will be good. Good luck on the new place.