The air damper was not operating so the cold air from the freezer could not get into the refrigerator. The part from AppliancePartsPros.com did work but not without a slight modification. I read one post related to the same part where the customer thought he received the wrong part because he couldn't get the clips to engage the freezer wall. The response from this site was that he received the correct part but he just had to push really hard.
Armed with this knowledge I was prepared to have to push hard to install the new air damper. However, even pushing to the point the Styrofoam enclosure on the part started to break the clips would not engage. At this point I compared the clip length on the replacement part to the original part and found that, in fact, the replacement part clips were approximately 3/32" shorter. No amount of pushing was going to make up the difference. I cut off roughly half of the foam gasket around the opening to make it thinner but still left enough to seal. This allowed the clips to reach through the freezer wall and connect properly.
Aside from that the installation was simple and effective. My refrigerator is now nice and cold as it should be. The information on this site was very helpful in determining what my problem was and how to test to make sure that was the problem before ordering the part. My advise to anyone ordering this part is to compare the clip length on the replacement with your original part as soon as you remove the old part. If the clip length is significantly different then modify the foam seal thickness to compensate so you don't risk destroying the replacement part in the process of installing it. I was able to repair my replacement part where the Styrofoam broke but others may not be so lucky.