Can the transfer case be damaged when a new transmission is installed?

The fluid does not need to be drained but if it was, did it not get refilled? Also, when you say it went out what seems to be the problem? Can you please shoot a quick video with your phone so I can hear the noise? That way I can tell for sure what's going on. You can upload it here with your response. Here are some diagrams in the images below to show you how to check the fluid and the component locations as well.

2 days after the transmission was installed, when decelerating from a rapid speed (like on a freeway exit ramp), the engine would race and the dash display would show an image with it flipping through High 4 WD, then Auto, the Low 4 WD. When you stopped or turned the engine off, it would be normal. It never occurred when the gas pedal was pushed and the engine engaged. I just found it suspect that this never occurred until the transmission was replaced. I know that it goes out with mileage, but seemed like a big coincidence? So I wondered if the transfer case could have been damaged when the new transmission was installed.

Yes, that is possible. Are there any codes in the car? This guide will help you gather them:

https://www.2carpros.com/articles/checking-a-service-engine-soon-or-check-engine-light-on-or-flashing

The code read is easy to do, let me know what you find.