Before you go looking for the hard stuff, charge the battery at a slow rate for an hour with a portable battery charger. Unplug the small connector on the side / rear of the generator, then drive the vehicle to see if the symptoms are different or cleared up.
If things are different, have the charging system tested for maximum full-load output current and "ripple" voltage. If current is exactly one third of what it should be and ripple voltage is high, the generator has a failed diode and will need to be replaced. The third potential symptom of a failed diode is elusive engine running problems. This generator design used by GM starting with 1987 models develops huge, harmful voltage spikes that can damage the internal voltage regulator, short one of the six diodes, and interfere with computer sensor signals. The secret is voltages can be "induced" into sensors' signal wires, and while the resulting voltages the Engine Computer sees are incorrect, they fall within the acceptable range, therefore no diagnostic fault codes are set. About half of the over 2000 potential fault codes will turn on the Check Engine light, so if there's no fault codes, there will be no Check Engine light either. The Engine Computer will accept the false sensor readings and try to run on them.
The most important part of my story is if the generator is found to be developing only one third of its rated maximum current and ripple voltage is high, you must replace the battery and generator at the same time to reduce the high number of repeat failures, ... Unless the battery is less than about two years old. Repeat generator failures on GM vehicles are very common, but it is mainly caused by those voltage spikes they develop, and those spikes are dampened and absorbed by the battery. Batteries lose their ability to do that as they age and the lead flakes off the plates. It might crank the engine just fine, but it's no longer able to absorb those harmful voltage spikes. If you don't replace the battery when this happens, it's common to go through four to six replacement generators in the life of the vehicle.
Saturday, May 14th, 2016 AT 11:11 AM