Can I extend my 60-day non-B visa while waiting for my work permit approval in Thailand?

You basically start over from the beginning. In immigration's mind you had enough time if it was a real job. 90 days is more than enough time to get your work permit, given you had enough documentation to get the type B to begin with. They can make extreme exceptions if there was a real ligitimate reason you couldn't get it, like your hospitalisation, and such. You need to go to immigration before it expires. At the worst they will give you a seven days to leave stamp, then you start over. Sorry, but immigration has so many people trying to game the system that they are much less tolerant than just a few years ago. You might be able to get a covid extension for 30, or 60 days, but most have been asked to see their exit ticket out of Thailand in that timeframe, but not always. This could give you enough time to get your work permit situation worked out, then get the new type B visa, and your work permit at the same time.