a dtv is a tourist visa and can only be obtained outside of Thailand. Keep in mind that a dtv visa will not work to open a bank account in Thailand and they have gotten very strict about that due to large scale money laundering. You can explore all the different visa types on the official Thailand evisa website thaievisa.go.th and find what works best for you but relocating with a family for extended period I would say the dtv visa is not going to be the best and safest option for you. A final thought to remember is once you enroll your kids in school they will need to obtain a education visa for the school to accept them into classes but most international schools in country can help you navigate that process
Is the cooking school in Thailand or Korea? Sorry the post is a little confusing. Was the course at least 6 months? Was the application itself rejected ( often due to a mistake in the form) or the visa denied ( often due to not meeting the requirements)??