Is it GoDaddy's DNS or is it cached browsers?

The answer is both.

At some point in the request cycle, after (or maybe before) my NginX webserver and Passenger app server processed it, the GoDaddy Apache server must have gotten a hold of the request object and added the .php or .htm file extension to the url.
Then, because of browser caching, the browser seemed to expect the extensions.

To correct this problem:
I changed the Apache .htaccess file to rewrite the URL without the extensions but that seemed to do nothing.
Then I took any php or htm files out of the GoDaddy root directory where Apache would be looking for the old site files. I figured Apache would just pass on whatever it had and not change it since it couldn’t find any files to process on it’s end.
After all that I cleared my browsers’ histories.

It seems to be solved.

No more .php or .htm and my administrative links are working as expected.

Intriguing to say the least, I thought if I pointed the site to my VPS ip address and pointed the nameservers to my VPS all GoDaddy assets would get bypassed. I guess not.

Wouldn’t you also think that GoDaddy servers would be bypassed?
I’m still curious as to how this was possible.