If they contain important info: select the messages you want to save, at the bottom of the message list is a button that says Selected Messages, click that button and it will give you the option to download them as text, csv, and a couple other formats.
They'll all be in one file but easy enough to scroll through or use the search function.
To clear up space quickly you can delete sent messages that have been responded to since that text is by default copied in the reply. Verify before deleting if its important.