Hello frog11,
Welcome to the Microsoft community.
We have also tested here, Japanese IME does open a new window with English input by default, and you need to switch to hiragana input once manually. After switching to hiragana input, you can continue to use hiragana input when you switch to Japanese IME again, even if you switch to another input method, as long as you do not switch windows. This seems to be a design problem, so we suggest you to give feedback through the feedback center.
Send feedback to Microsoft with the Feedback Hub app - Microsoft Support
However, it is not normal to enter "ta" and have it recognized as "tあ".
Have you tried to reinstall the Japanese language pack? If not, please try it to see if it can be restored.
Open Start>Settings>Time and Language>Language and Region, find Japanese and uninstall it, then add Japanese again.
If the above methods do not solve the problem, we suggest you use the previous version of Microsoft IME and see if the problem still occurs.
Right-click the ”A/あ” in the bottom right corner of your desktop , select Settings > General, and under Compatibility, turn on Use previous version of Microsoft IME.
If there's anything that's not clear, don't hesitate to let me know.
Sincerely
Carlos - MSFT | Microsoft Community Support Specialisto