Maybe, but i kept a scanned copy of the signed form letter I was required to submit stating that I was leaving the job. And my Thai government university workplace took the letter and my passport and work permit book to immigration and later declared to me that it had been handled. As mentioned above, we did this prior to the expiration of my most recent extension, and I left Thailand before the expiration of the extension, without renewing. I might come back someday with the intention of retiring in Thailand for a while, so I guess I'll learn more then.

If the company says to come with them, then do so. In my case, it turned out a bit different, as I was working with an annually renewable non-B visa and then after some years stated my intention to leave one month before my anticipated departure from Thailand. At first the administrative staff said I would need to come with them to the immigration office to cut my extension short (about 1 months before annual renewal), and I filled out a form stating my intention to leave before the end of the contract and gave the staff my passport, but then they took care of it all at the immigration office without my needing to come with them or do anything else. I worried that maybe something had been missed since I didn't personally come to the immigration office, and so maybe they would fine me, but when I came through Thailand six months later as a visa-exempt tourist, they admitted me through immigration with a smile and no issues. But it seems like employees may normally need to come to the immigration office.