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Kazakhstan does have exit visas but not for this situation. (see below) The document that could cause an issue exiting the country is the loss of or a failure to properly register a migration card. Unless this is a very new thing then the current employer is wrong. I have been here a while and had several visas and also changed employers.Your are correct about the process, end or terminate contract, leave country and get new visa



Exit visa (ШЖВ)

Exit visas are issued to:

Foreigners permanently residing in the Republic of Kazakhstan  on the basis of the permission of internal affairs authorities (Immigration Police);
foreign nationals who have lost their passport or other documents granting the right to cross the state border of the Republic of Kazakhstan while in the territory of the Republic of Kazakhstan - on the basis of written applications of diplomatic missions, consular institutions of foreign countries and certificates of return to their country of permanent residence;
Aliens subject to deportation from the Republic of Kazakhstan in accordance with a proper court order.

I was able to leave KZ on Saturday without difficulty and no exit visa or other such document was necessary. Now, to change the subject slightly, getting a visa to come back to KZ. Here's what I just received from the employer that is trying to hire me:

"I gave your documents and certificates with diplomas to the ministry of foreign affairs [sic]. Unfortunatelly [sic], they said that they do not provide visa to holders of TESL or Celta [sic]."

So, no more native speaker English teachers who have TESOL certifications (e.g. CELTA)? Something seems strange about that. Before I left, the prospective employer said that the Ministry of Foreign Affairs had said something about wanting to see on the university transcripts something about being eligible to teach (something you will not find on American university transcripts since eligibility to teach comes from state governments, not universities).