In case anyone happens to somehow have the same problem I did, I have finally found the solution. It seems it was actually an issue with Windows and not Unreal Engine. How I found it is a long story, but it turned out apparently I was missing the cmd.exe program in my system 32 folder (it’s possible there were other issues that were also fixed by my solution, but this is the one thing I noticed). I have no idea how this happened. To fix it, I essentially just followed the guide from microsft support found here: Use the System File Checker tool to repair missing or corrupted system files - Microsoft Support
I ran the DISM tool for RestoreHealth, and I also ran sfc /scannow from the command prompt. After doing both of these things I checked and found that my cmd.exe was back in system 32 and I could package my games right away.