Can I stay in Thailand after my work visa is cancelled and what is the process for obtaining a new Non-B visa?

I am currently in Thailand on a Non-B Visa and recently received my Work permit. I am still in the initial 3 months period of the original Non-B visa.

Another school offered me a much better job (Salary and environment) which I accepted so I will resign from the current school within the next day or so as I am really not happy there. I would like to stay in Thailand until I start the new job in May. My current Non-B visa expires on the 15th of May.

1) Can I stay in Thailand after my work visa is cancelled until the Non-B, 3 months period is over or do I have to leave Thailand immediately? My current employer will definitely cancel my work permit immediately as they have done it with other teachers.

2) I know I have to get a new Non-B for the new school as the name of the current school is written on my Non-B visa. Can I go to Laos to apply for a new Non-B visa directly or will I have to do a border run first to apply for the new Non-B as a tourist?

Thank you.