What options do I have for renewing my Non-O visa for my Thai children while traveling?

You better go to your local office and talk about what you can do.

Bring fixed air tickets (an open ticket wouldn't do the trick) as proof you are not in Thailand during the application window.

- For 1-year extension application:

Some offices may allow you to apply for 1-year extension in advance when you can prove you are out of Thailand.

- For 60 days family extension:

Some offices add 60 days to your current stamp, while some set 60 days starting on the day of application.

If both are not working well for you, you have to give up the current extension. You do as you explained, or you can apply for an initial Non-O visa at the Thai embassy/consulate before you will be back.