I upgraded and a screen came up that required me to login. My Password didn't work and when I went to Password Reset that didn't work either an error message came up that said "Mail send failed. There is no mail handler." This must be a bug in the upgrade login.

I need a password reset ASAP as my client can not work until we can login, finish the upgrade and gain access.

No bug, if you did not enable one of the two mail plugins (which depends on the server) and have configured them correctly plus the admin account requiring the reset has not set a valid e-mail it can't work.

For a manual reset please see:
http://www.zenphoto.org/2009/03/troubleshooting-zenphoto/#16

However, I am receiving the link to reset going to admin@zoesternberg.com but when I go to it sends me back to the upgrade login page and then the CAPTCHA does not work.

Ok. Standard response to this question:

"What version of Zenphoto are you running?"
"How was Zenphoto installed?"

Version 1.3.0 through Install Central with Fatcow Web Hosting. When I tried to upgrade to 1.3.1.2 I got into a loop which I haven't been able to get out of.

Sorry, we can't support install helper scripts which I assume Install Central is. This is both noted on the forum rules and our installation page because we get frequently reports of upgrade failing.

Please do a manual install as described on our user guide or contact the vendors of Install Central.

Maybe I should cut my losses and just uninstall and re-install ZenPhoto . . . I just want to be sure that the software is stable enough for my clients.

It probably is if installed correctly. But we continue to have problems when users have used some unknown tool to do the install. We of course have no clue as to what those are doing so do not know what needs to be corrected. Installing is pretty simple. I suggest that you try a clean install using the zenphoto instructions.