Hotkeys such as next/previous tab affect not the current window, but the original window. So if you open a shell and pull its tab out, you'll get a new window. If you do some more, you may have multiple windows, each with multiple tabs, as you'd expect. If you now think that hotkeys will the current top window, you're wrong - e.g. changing tabs back and forth will not change those tabs of your current, but those of the original window.
I've just released a new beta version which should fix these issues.
As usual, you can get the beta here: https://www.royalapplications.com/go/kb-ts-mac-downloadbeta
Please let me know if it fixes the reported issues!