When I drag drop a message to another application it is moved to “Deleted Items”

Some applications request a move action when a file is dropped. By default Outlook honors this request and the item is moved to “deleted items”.
DragDrop for Outlook prevents this behavior by forcing a copy action when dragging to a well known browser.
For version 5.5 this list is:
chrome.exe; firefox.exe; iexplorer.exe; microsoftedge.exe; opera.exe; chromium.exe; msedge.exe

When using the Enterprise version you can extent or override this list in the registry: https://www.dragdrop.com/registry-settings/

If you are dragging to one of the supported applications, please make sure the DragDrop add-in is loaded:
https://www.dragdrop.com/dragdrop-setup-helper/ (step 1)