Personally I can't live without Vimium as it lets me do so much more than just move tabs.
You can easily open a tab into a new window with W or you can search across all your open tabs (in all windows) with T or open a url from your history with O. If you were editing text in an input field you may need to hit Esc first to "unselect" the field, but that is common with "modal" editors like Vim which Vimium borrows many of its ideas and key mappings from.
A new tab is just a t away as well.
Sadly it doesn't have any direct "send tab to a specific window" options like Tab to Window (or at least the version I used to use on Firefox allowed that), but it sounds like you don't need that.
For actually moving tabs to organize them I use the extension Tabs Outliner which works great and lets you "tag" windows and tabs with notes, and the BEST feature is that you can simply hit Ctrl+s to save out the entire list of tabs to send via email or backup with Dropbox/Drive/etc.