What you ask for, "actively correct spelling" and "grammar check", are not possible with Mozilla spellers.
For that you need Microsoft Word or OpenOffice/LibreOffice Writer, because they have an option called "autocorrect" which allows to replace words automatically (I am working on it for LibreOffice in Portuguese).
The best you can do is to use LanguageTool with Firefox:
Notice that the add-on for Thunderbird is kind of unmaintained.
Also, have in attention that my GB speller is the official one for Mozilla:
I am not sure if they are shipping it with the software though, which means that you probably have it built-in without knowing.