If both fans come on, unless the fans are failing, that is because the A/C compressor has been activated.
Any whirring, scraping, or clicking noises from the dash? A blend door actuator could be at fault.
Sounds more like a bad battery, or that the HVAC module needs to be reset. If the battery (verify CCAs) checks out, then I would pull the neg cable for 40+ minutes and reconnect. See if that makes a difference.