I want to change a unit name, but the name is too long, so it is partly cut off. Can I still change it to the full name somehow?

If your name is too long in game then that's the way it is.
If your name is too long in the unit properties then change it in the string section.

If there is anything that you can't change in the Strings, you can always change them using Artmoney.

I can change the words, but how can it be saved? After closing string editor, the text is changed back again.

click on a different string to save the string.