Not a mod, but I’m trying to help as many people as I can. To help Mojang fix this, I’ll try to replicate your issue if this doesn’t help fix your issue.
So you can’t select any skins or add your own? Try logging out of the Minecraft Launcher with your Xbox Account. And sign back in. This should refresh your skin files, if this doesn’t work, a way to get around this is going to Minecraft.net to go to your profile. Go to account, click on profile. Go to all games. Click on change skin through Minecraft Java edition. See if that works as well. If not, it’s probably just an error with your account. Let me know how it goes.
Quick question to help replicate this issue. Did you put in un-stackable items into your Shulker Box? Also to help out with reporting this bug. Are you on an original Switch? Like a launch model? Or Lite, OLED, or HAC-002 and onwards. Or a Switch 2? Let me know and I’ll try to replicate this issue to help Mojang find the root cause of this.
Shoot, I would think the next step of troubleshooting would be messing with how to install the launcher. Like doing a manual download from Minecraft’s official website, or downloading the launcher from the Microsoft Store. Another thing to check would be messing with the run with admin on your desktop on the Minecraft Launcher. Hope those help if you haven’t tried those yet. I don’t have any other ideas other than a clean wipe on your hard dive, maybe something in the background is causing this.