Can I switch jobs in Thailand with a Non-imm B visa if my work permit shows a different employer?

Hello, hoping someone can advise.

I have arrived in Thailand with a Non-imm B for teaching. My job (i.e. school where I'll teach) is not finalized yet - the agency prepared the docs so on paper I am working directly for them - not signed contract yet. No employer specified on the visa sticker.

Now, I have been offered a separate school job unexpectedly. Decent school but zero idea how good their paperwork people are re. Ministry of Labor etc.

As you can guess I would like to take the job offer, but my WP3 papers show the agency job. DO I have any options? Is there a legal or legally accepted way to take the job offer with the existing visa?

Thanks in advance and hope I included everything relevant.