IntelRobert - okay, so I used the aapt tool. The APK I uploaded to Google Play was version 20016. When I built a new APK on my 2nd computer, the version number was only 20010. So, right now, I am doing 7 builds in quick succession to get the version code to be 20017 to see if that works.
***Edit*** Okay, yes, I can confirm that my 7 quick builds in a row got the version code number high enough to be accepted.
In case anyone wants to check their version code using the "aapt" tool, the correct syntax in Windows command prompt:
aapt d badging your_app_name.apk