Hi John,
There seems to be 2 questions. Question 1 is the issue of activation for Office Pro Plus 2016, Question 2 is about how to upgrade Office from Office 2016 to Office 2019.
Question 1, please search CMD, right-click the Command Prompt and run as administrator. Then copy and paste the command, click Enter at last:
cscript.exe "%ProgramFiles%\Microsoft Office\Office16\ospp.vbs" /dstatus
If you are running 32-bit Office on a 64-bit operating system, please run:
cscript.exe "%ProgramFiles(x86)%\Microsoft Office\Office16\ospp.vbs" /dstatus
And then, please copy the results or make screenshot to me, I will check the activation status first.
Question 2, Office 2016 is a one-time purchase (non-subscription) version of Office, to upgrade, you need to buy the newer version. To confirm the requirements of Office license based on your conditions, it's best to contact Microsoft for help, as we focuses more on general questions related to Office desktop applications. Thanks for your understanding.
To deploy it on machines, you may use ODT tool with Config.xml file. As machines are all connected to internet, you may get installation files from CDN (Office Content Delivery Network) directly.
Besides, you could try to add "RemoveMSI element" element in Config.xml to remove previous Office Pro Plus 2016.
Here is the article about Office 2019 deployment, "Deploy Office 2019 (for IT Pros)", I think it it helpful.
If you have volume licensed versions of Office 2019, such as Office Standard 2019 and Office Pro Plus 2019, you may use MAK activation which activate Office through Microsoft-hosted activation services, either via the internet or by telephone.
In this case users could use Office apps without logging their Microsoft Accounts.
Any questions please post back.