IGNORE REPLY -- THIS IS FOR THE SEARCH BAR, NOT THE FIND BAR

Firefox's search bar may display suggestions from your default search site or from past searches. You can disable search engine suggestions using the search preferences. For your own search history, read on.

By default, Firefox stores form history and search bar history, and these are managed together as a single feature. To disable that:

• Mac: "3-bar" menu button (or Firefox menu) > Preferences
• Windows: "3-bar" menu button (or Tools menu) > Options
• Linux: "3-bar" menu button (or Edit menu) > Preferences

In the left column, click Privacy. Then on the right side, change the selector to "Firefox will: Use custom settings for history" if it doesn't say that already.

Then you can uncheck the box for "Remember search and form history"

(There's no save button, the change is saved right away.)

Does that stop Firefox from showing search history on the search bar drop-down?

Turning off this feature doesn't erase old entries. To do that most efficiently, you may want to try the following add-on:

https://addons.mozilla.org/firefox/addon/form-history-control/

Whoops, I was thinking of the Search bar, obviously, not the Find bar, so please ignore that first reply.

It's a little unpredictable to me what will appear in the Find bar when I open it. Tabs opened in the same window AFTER I have used the bar, often have the last-searched-text in the bar. However, if text is selected in the page, then the Find bar will contain the selected text. I don't know of a way to bypass either of those behaviors.

A very comprehensive add-on you could look into is the following, but the description doesn't mention your specific question:

https://addons.mozilla.org/firefox/addon/findbar-tweak/