Your ability to create, copy, move folders and files depends on several things. I don’t know all the rules, but here are few stabs in the dim.
The folder(+) icon lets you create a new folder. This is now standard when you need to save, move, or copy files; hurray.
In general, you can’t move folder outside an app. You can only Copy them. You’ll see a Move option when in Select mode, but when the folder destination chooser (or whatever it’s called) pops up you may see Copy instead of Move. By experiment I can move folders easily within my main Documents by Readel iCloud folder, but when I try to move them outside the folder I can only copy them.
This probably has something to do with underlying folder/app permission/sandboxing in iOS.
There may be further problems with apps that have not been updated for iOS 13 (especially other cloud services like Dropbox), or apps that are using local folders. I rarely use local folders these days. It’s easier to keep all content available on all devices if you use the cloud. Just keep in mind whether you are in a local folder, iCloud folder, or some other cloud service. Maybe it will make more sense then.
It is now possible to create folder that are not associated with an app, both locally and in iCloud. Just create them in the top level of iCloud or On my iPad. Moving and copying to from these folders may be easier.
I’ve been using Documents by Readle as my main file system. This is possible because Documents acts like a file server, letting other apps access. With iOS 13, that’s not necessary. You can create a folder structure independent of any app, and most apps will be able to access and copy/save/edit those files.
I haven’t done this, yet. I may later when I’ve done some more tests and catch the re-organize bug.
If you show me a screen shot of what is happening (with any personal data removed) I might be able to guess what is going on. Otherwise, these random thoughts are all I can offer. At least for now.