What should a university lecturer with a non-B visa do if they resign and wish to work at another university in Thailand?

When working and holding a non-B extension, that extension becomes null the moment the reason for that extension changes. So if someone quits or is fired, their permission to stay in Thailand is over that same day, whether or not the still have an extension in their passport.

It is that person's responsibility, based on the documents they signed to get their extension, to go to immigration and cancel it.

Once it's cancelled they either need to leave the country the same day, or apply for another extension they are eligible to receive.

If someone wants to change jobs without leaving Thailand, they would need to coordinate with their old job, their new job, and immigration, in order to get everything arranged before the end date of the first job and then go and then get the new work permit and go and cancel the Non-B extension and apply for a new one.

Else you would get the documents from the new job, cancel the extension at immigration, and go to a neighboring country and apply for a new Non-B visa.

If you don't do this, you are on overstay as of the day after the job ends. Not a good situation for anyone.

I hope you're not teaching English