Entering visa exempt or tourist visa is a catch 22.
You need a Thai bank account in your name only to obtain a non O.
The bank wants you have a non O to open bank account.
As I posted above you may be able to obtain certificate of residence from immigration in your province.
Alternatively your embassy can provide equivalent however some embassies do not.
USA, UK, AU to name 3. Agents can assist with opening bank account however even this has become difficult in Bangkok.