2003 Ford Escape died today, would start with a jump but not stay started. Replaced battery and went to part store, hooked up to their machine which said it was putting out 12.13 volts. The clerk stated my alternator "is starting to go out" I thought it either worked or it didn't. Is it charging my battery? Car safe to drive? Do I need to replace the alternator right away?

Sounds like it's not charging. Voltage should be above battery voltage. Anywhere from 13.5 to 14.5V
Have the alt replaced. Fully recharge the battery.

The alternator is bad, since the regulated battery voltage is too low. This link may help:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-hondas-selected-other-vehicles-46563-10.html#post1973613