You can rebuild the AAM-7, and a situation like yours is the only time I would do so. If its just worn or needs a fresh-up...its worth putting a couple hundred bucks into it. However, if its been blown or broke...I would just throw it away. Its normal to have quite a bit of lash. All of the AAM-7's I've seen have been relatively "loose" compared to most other production axles I've seen. That's not cause for alarm, unless you can physically feel some "shake" in the pinion bearing or it is whining at highway speeds (usually during coasting).
If you heard popping, and checked the front diff and found nothing....then that wasn't the problem. It was probably the t-case binding (sticking between modes), a CV-joint, or one of many other components. The entire suspension on the H3 can make all sorts of noises and this makes it hard to troubleshoot the real problem. Frankly...I'de fill it up with fresh lube and put it back in. Look elsewhere for the noise.