The first thing to consider is you already addressed a real common problem on GM vehicles by replacing the battery. Since the charging system was redesigned for the 1987 models, GM has had a real big problem with their generators developing huge voltage spikes. Those spikes can damage the internal diodes and voltage regulator, and interfere with computer sensor signals. The battery is the key component in damping and absorbing those spikes, but as they age and the lead flakes off the plates, they lose their ability to do that. It is common to go through four to six replacement generators in the life of the vehicle unless the battery is also replaced.
To reduce that number of repeat failures, always replace the battery at the same time, unless it is less than about two years old.
A common cause of back firing and other elusive running problems is a failed diode inside the generator. One of the three output phases will be missing, and that results in "ripple" voltage being very high. That adds to the interference with computer sensor signals. It also makes the voltage spikes more pronounced and harder to dampen.
Before you go looking for other causes of the running problem. Unplug the small plug at the rear side of the generator, then take a short test-drive. If the back firing is gone, have the new generator professionally tested by a mechanic with a load tester, not at an auto parts store. If a diode has failed, the most you'll be able to get for the "full-load output current" is exactly one third of the generator's rated current. Charging voltage may be okay, but 30 amps from the common 90-amp generator is barely enough to meet the demands of the entire electrical system under all conditions.
If the running problems are still there with the generator disabled, you will have to continue with the normal diagnostic steps.
Wednesday, March 15th, 2017 AT 5:44 PM