Firstly, what I learned from Google is that there is a daily data limit for gmail accessed through the IMAP protocol. IMAP protocol - which is what Apple Mail uses to access gmail accounts - asks for the entire gmail account to be updated both to the gmail server and to the computer/iOS device. I believe this means if you have a iCloud file structure into which you sort emails, IMAP is looking everywhere for changes and communicating that - it's not just communicating new data, according to the Google tech I spoke with. When the daily data limit is hit, the transfer slows waaaaay down or stops at least temporarily. Compounding the problem is that each device connecting via IMAP eats into that daily limit - the more devices you use via IMAP, the more compounding of the problem.
Partial Solution #1:
What I finally did was go into gmail console from web browser and create app passwords for my two iOS devices. Then used those passwords to set up my gmail account as an Exchange account, then deleted the old gmail account that uses IMAP protocol. It is not possible to set up gmail using exchange on MacOS. We'll see if this solves the problem. So far so good.
Partial Solution #2: What may have made the data limit visible was something that seems to have begun last month with the upgrade to 10.13.4 - a problem may have occurred with the mac mail program. The symptom of the problem was that every time I'd log onto my computer and open Mail, it would start downloading thousands of emails. While I couldn't be sure that these were only gmail emails, it is likely that they were, and each time it was eating into the data limit. I worked with a senior tech at apple and may have solved the problem by deleting one particular mail file in the user library - but I would suggest a call to apple for guidance on that.
So setting up my iOS devices using Exchange and by solving the problem with Mac Mail, I think I brought my daily gmail data use way way down and so hopefully will not see the problem again.