Quit the agency and get a proper job, there are plenty of schools looking and they will help you take care of your WP and visa. Agencies in Thailand are sharks and scammers generally, but not all. Before COVID you had to have a non-O to transfer for your WP, so if in a tourist visa you need a letter offering you employment which you would then take to the embassy or consulate outside Thailand to exchange for a 90 day non-O visa, which will allow you to start your employment while the school arranges your WP and visa. This may all be doable within Thailand with the current COVID situation, but you will need to check that. So my advise 1) get a letter offering you employment 2) exchange your visa for a non-O 3) start work and the school should apply for your WP and visa.
I've looked on the website already hence my confusion. I cannot visit immigration because I'm outside Bankok now. What form do I need to complete and can I print one? I cannot find any printable forms on the website.