Sure sounds like a power supply problem. You could have a bad alt. bad battery or a parasitic drain. You could also have a bad ground. If it were my truck, I'd leave it sit for 24 hours, and then check the key off voltage. Should be north of 12 volts. Charging voltage should be at least 13.5V. I would remove and clean the engine ground by the diagnostic plug (aka the 'sensor ground') and the ground to driver's fenderwall. Also check your battery cables for any bubbling insulation (sign of corrosion underneath). Clean and reattach battery terminals. Rule out a parasitic drain by letting truck sit a half hour or more, remove negative terminal, and measure current between terminal and battery post. Should be no more than about 50mAmps. A load test (the one the parts stores do) is the gold standard for battery testing but not really required for this symptom set.
It's no surprise that the ECU would be more sensitive to a power issue than the radio.
If all that doesn't do it, it's time to consider the ECU. You can't really test it other than by swapping in another one.