Q: How can I save messages without using up storage?

If you store them on the phone they will use storage space, its like your purse, put something in it and it uses space.

You'll need a 3rd party app. iOS has no native ability to do this outside of an iTunes or Icloud backup.

On your phone go to Settings iCloud and press on the word storage, then Change Storage plan

You could always capture them in screen shots on the iPhone, then save the screen shots on your computer.  Then delete the screen shots and the messages from the iPhone.