Just notice that my email yesterday did not include the body of the text for some reason. So, this is what I did to make the phones work:
OK, after trying everything I could find in the help sections which included everything from checking all of the network settings for the Republic and Motorola apps to resetting the Republic and Motorola apps; it looks like the final solution for me was to factory reset both phones. Took a couple hours each to complete - I discovered the phones had not been backing up to the gmail accounts properly so I had a bit of rebuilding to do but both phones seem to be working on the cell network as well as the wireless networks today with the latest version of the Republic app for the MotoG 1.
A couple works of advice that I am sure Support would have give to me if I had asked:
Backup all of your personal items locally (to a flash drive or laptop) before factory resetting the phone. The usual stuff, photos, books, downloads etc… - Factory reset will wipe all data from the user area of the phone.
Check the date on the backup in the gmail account to see how current it is. In android 5.1, I did not see a way to force a backup to be made. Newer versions may vary.
Check the contacts list in the gmail account against the one in the phone in case the backup is not completely current.
Thanks to all for the tips and links, it is all OK in the end but was pretty frustrating for a while there.
Salutations and wonderful days to all!