What should I do if my Non-O visa is expiring and I need to extend my stay in Thailand?

You wouldn't apply for a new visa in this situation, you would apply for a 12 month extension of stay at your local immigration office. But you need to meet the requirements of that immigration office including having 400k baht in a Thai bank

If you don't have the money in your Thai bank account that you need to apply for the 1-year extension, then your only option is to leave Thailand and either return visa exempt if you are from an eligible country, or apply for a Thai visa in another country. Most of the Thai embassies in nearby countries are advising a 10-15 business day application time. You can check each embassy's website for their visa requirements and processing times.