Can I switch from a tourist visa to a student visa in Thailand, or must I leave the country?

You can get your school's assistance in converting a tourist visa to an education visa. Ask them and they will advise you accordingly.

Yes no problem..

You need to ask your school. It takes hundreds of pages of paperwork to do an in country conversion and is only possible if your school supports it and provides the paperwork and if your immigration office allows it. Not all immigration offices allow it but your school should know if their immigration office does.

Also you need 1-2 months left on your stay or the ability to apply for an extension to get that amount of time because your school will need to apply to the ministry of education which can take 3-4 weeks and then you still need to have 14-21 days left on your stamp when you apply at immigration.