I've used the Clear Search 2 add-on before. It does exactly what it says it will do!

You can disable saving form data if you do not want to save search terms entered in the search bar on the Navigation Toolbar.

• Tools > Options > Privacy > Firefox will: "Use custom settings for history": "Remember search and form history"
• https://support.mozilla.org/kb/Form+autocomplete

Note that the search bars on the about:home and about:newtab pages use autocomplete=off and thus do not save this data.

I do want to remember form history, just not search bar history.

To solve your problem go to options -> on the popup privacy settings and then in HISTORY FIREFOX WILL: change it has never remember history

I only want it to not remember search bar history. Doing what you suggest will make it not remember every kind of thing.