What you are describing has also been reported by other macOS 27 users.
In particular, there are reports of dragging not working properly in Icon View, including on the Desktop, while the same drag works in List, Column, or Gallery View. (reddit.com)
The fact that dragging works for you in List View is an important clue.
It makes your problem consistent with the Finder behaviour others are seeing after the macOS 27 update.
Apple confirms that Finder has separate Icon, List, Column, and Gallery views. (support.apple.com)
Make sure you are running the latest available macOS 27 update.
The latest version of macOS is 27.0.1. Learn how to update the software on your Mac and how to allow important background updates.
Apple’s release notes do mention a Drag and Drop fix, but that relates to a developer API and does not appear to address this Finder/Desktop problem. (gist.github.com)
For now, the simplest workaround is to use List, Column, or Gallery View when you need to drag files.
If dragging suddenly stops working, relaunching Finder may temporarily bring it back; some users have also reported that resizing the Finder window helps.
Neither is a permanent fix. (reddit.com)
The scrolling issue while dragging appears to be a separate problem.
Other users have reported that, in macOS 27, they can no longer scroll a Finder window normally while holding an item during a drag. (forums.macrumors.com)
I would not reinstall macOS or start changing trackpad settings based on these symptoms alone.
If the problem continues after updating and relaunching Finder, report it to Apple through Feedback and include the useful details: dragging fails in Icon View and on the Desktop but works in List View.
Unfortunately, there is not a confirmed “Ultimate Fix” for this particular Finder/Desktop behaviour yet.
For now, changing Finder views is the most practical workaround while waiting for a macOS update that addresses it.
Apple: Organize your files in Finder