I've been organizing my wardrobe and changes to drawers from the drop-down are saving as I go. However, maybe it's due to a delay in the server response but there have been a few times now where I've tried saving some items to drawers and it takes 4-5 attempts to get it to save to the correct drawer. In order to get it to work, I've been refreshing, opening a new page, trying to change the page/drawer without exiting the organization page of the wardrobe, among other things but it's been a bit glitchy.
I'm on Windows 11 Home, using Chrome browser.
This has been a long-time issue for me, as well. The automatic saving is not reliable. When it works it's fine, but often I will have to sit on the same page for extra time, or will have to refresh and try repeatedly as the OP has mentioned to get it to work. Sometimes I just have to leave and try again later.
Personally I would prefer going back to having a save button.
MX 19.4 x64 (Debian-based Linux) with Cinnamon desktop Vivaldi browser (Chrome-based)

I'm using windows 11 as well, Firefox browser, that has always been an issue for me with drawers which is why I try not to use them. They're so wonky ;u;
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Have been encountering this issue too
EDIT: never mind, when I came back in a few minutes it had worked - just a bit slow
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i also find that if i accidentally select the wrong drawer, and then switch to the correct one, it only saves the first selection so i have to refresh the page and do it again. wish there was some sort of visual feedback to confirm that the drawer has been updated.
When I click the delete drawer button, there is no confirmation. It simply remains in the list until I refresh the entire page. There is no confirm button for updating what drawer an item is- considering there is also no mass select item options either, this is a tedious enough process without also having to just… hope? It saves any change I make. It also often doesn’t even save the change, I can refresh and check items with no drawer, and the same whole page of items I tried to sort is still saying they’re I sorted.
This is consistent across iPhone, macbook, and a windows pc, with the browsers chrome safari and Firefox tested on each device.