See this - https://support.mozilla.com/en-US/kb/Websites+or+add-ons+incorrectly+report+incompatible+browser

I inactivated all my add-ons. They are not the problem. But I don't want to mess with the user-agent. Any other ideas?

I can't believe Mozilla can't recognize it's own version of FF!!

You can find the real Firefox version in the Help>About Firefox screen.
https://support.mozilla.com/en-US/kb/Finding%20your%20Firefox%20version
At this moment you are telling everyone that you have version 3.5.1, and that is what we recognize on this forum. We have no reason to assume that this is wrong.
Like the-edmeister said, it is possible that you useragent information is wrong and needs to be corrected.

"About Firefox" on my computer verifies that I am using 7.0.1.

Your Firefox UserAgent is messed up and is telling websites you visit that you are using Firefox 3.5.1.
Mozilla/5.0 (Macintosh; U; Intel Mac OS X 10.5; en-US; rv:1.9.1.1) Gecko/20090715 Firefox/3.5.1

Reset the prefs related to the UserAgent as described in the article that I hyperlinked in my first answer.

I tried to make the mods, but Reset for general.useragent.locale is greyed out. The other options for useragent, don't appear.

Put this string - general.useragent. - in the Filter line at the top so you can see all the prefs related to the UA setting all at once. Reset any that are bold. Prefs that aren't displayed in a bold type are already at their default setting and don't need to be reset.

That worked! Thanks for your help.