2004 Chevy Trailblazer, battery dies but engine stays running & other weird electrical issues

If the engine is running and the headlights stay on then it's unlikely you have a problem with your battery or your alternator which charges it as they must have power to run. All the dash lights, gauges and some auxiliary systems like door locks are run by computer, there's an engine computer, a body control computer and an instrument panel controller, if they lose connectivity to each other or lose power then all sorts of strange stuff can happen.

You most likely have:

• A grounding problem where one of the controllers or computers loses its electrical circut
• A connectivity problem between your computers or controllers
• A fault with one of the computers or controllers, i.e. one of them has broken

It's possible that your car may be able to tell you when the problem is if you get the On-Board Diagnostics (OBD) codes read. Almost every car past a certain year has one, and the codes can be very handy in figuring out what's wrong with your car. In the US most auto parts stores will read your codes for free, and every professional mechanic should have a reader. So get the codes read, and depending on what is found it may help. You can update your question with them if you like.