I don't wish to completely disable the 'Auto-Complete' feature. I want to be able to quickly access websites previously visited by me, instead of having to type the whole name, but I also don't want Firefox to show the contents of what I visited on that particular webpage in the Address bar.
Tick "Always clear my private data when I close Firefox"
Hit the Select button next to it and tick "Saved Form and Search History", (un)tick anything else you want or don't want.
Then OK you way out.
Or, Tools->Options->Privacy and Untick "Remember what I enter in forms and search bar".
Should be one or the other - if I read you correctly, the thing you don't want it to clear is your Browsing History.
AFAIK, Firefox has always had the ability to do this, (at least from v1.2 IIRC), but it might have been reset by an update.
I think, however, that if you have a user/password saved for a site it will override that setting so you can log in without having to enter your details.