I want to delete documents off of desktop, but not out of Finder

You can ask Finder to open an additional Window WITHOUT closing the other one.

File > New Finder Window

This can allow you to drag and drop from one window to another.

When you drag-and-drop between windows on the SAME drive, the file will be Moved (by changing its pointers in the directory, NOT by copying followed by deleting)..

When you drag-and-drop between windows on different Drives, the file will be duplicated at the new location by default. The original will remain.

All you need to do is drag and drop them into another folder not on your desktop. Maybe your Documents folder. Then send to Trash the ones left on the desktop

Finder is a sort of a window into your computer. Removing it from Finder is same as trashing it. What you do is move files to the places where you want to see them. Unless you hide them, they will always be visible by Finder somewhere. That's what finder does among other things ... lets you see files and folders all over your Mac.

"The Desktop" is a special way of presenting the contents of one magic folder, whose name is exactly Desktop.

if you would like to view those files sorted in a different way, you can open that folder is your Userid named Desktop, select the presentation by list, by tiered list, or other. in list view, you can sort by name, by date, or any other attribute by clicking on the column heading.