If you obtain a legal job and a residence visa, you will be able to obtain a residence visa for your children. However, most entry-level TEFL jobs do not pay enough to enable single parents to support children. Married couples in which both partners teach EFL may be able to support children. For most countries, it is usually possible to arrange a job in advance, particularly if you have a degree and a reputable TEFL certificate. Unless you are looking for a very short term job, I would strongly recommend that you fix up a job in advance.
Some people travel to their target country to look for work.
Others arrange a job in advance by answering job advertisements or by contacting possible employers. You can find some useful tips on finding overseas jobs on our TEFL Job Search Tips page. You can find more detailed advice, together with hundreds of addresses of employers, in the Teaching English Overseas book. Accept a job only if it is with an organization which has an established reputation or which is known to someone whose opinion you trust. Check out what other people say about the employer on Internet discussion sites and ask the employer to put you in touch with a current or past employee.