Since Claws Mail 3.11.0, new mails in the drafts folder are marked new to raise attention to unfinished mail. However, the trayicon behaves like when you have new _incoming_ mail, i.e. the icon for new incoming mail is showed (the white envelope in the gray mailbox), which is misleading. I am using Claws Mail 3.11.1 on a Debian installation.
The plugin ignores draft folders and their subfolders. Make sure your draft folders actually have the "Drafts" folder type, or are subfolders of a folder of this type.