In the headliner above the driver there is a little round plastic piece that looks like it should be a speaker... But it is an ambient temp sensor. Mine was missing and my truck was doing exactly what yours is. Put that sensor in and everything worked like it should instantly . It reads the inside air temp to calibrate the climate control with the reading it receives from the outside temp sensor that's up in the front grill/bumper.

 

Here’s a simple answer you can use:

It sounds like the issue might be with the blend door actuator on the driver’s side. The blend door actuator controls whether hot or cold air comes out of the vents. When it fails, it can get stuck, which is why you’re only getting hot air on that side.

Replacing the actuator is a common fix and something you can do yourself if you're handy with tools. It’s usually located under the dashboard on the driver’s side. You can find videos online to guide you.

Since you replaced the battery recently, it’s also possible the HVAC system needs to be recalibrated. Here’s a quick way to try that:

Turn off the truck and remove the key.
Disconnect the negative battery cable for about 10-15 minutes.
Reconnect the battery and start the truck.
Let the HVAC system reset itself (don’t touch any controls for a minute or two).
If this doesn’t help, or if you’re not comfortable with DIY repairs, you might need a mechanic to look at it. But checking the actuator first is a good place to start.