I also got out of a hagwon contract and jumped to a public school job. I did it, though, in the eleventh month of my second contract after I knew I was not going to be renewed by that hagwon and a student of mine told me that the public schools in that city were hiring. Like YBS, I had to force my way out. I didn't call the parents, but I caught that hagwon director telling one story to me and a totally different one to my wife. I also gave up around three to four million won in end of contract bonus, last month's salary, etc.
I also suggest that you don't try to secretly change your visa to that other job without properly going through the transfer process with your employer and the other school. You will get caught with your trousers down by Immigration eventually. They won't be very sympathetic with someone who's trying to get out of their contract after only a couple of months. Do it the right way and you won't have to worry.
It might be better for you to wait longer and then start the process of getting out. Frankly, it doesn't look that good. It looks better that way, too, on your EFL employment history.
And if you have not taught before, you can learn a lot about how to teach in a hagwon, because when you do jump to a public school job, most likely you will not have any where's near the support you do at your hagwon. Plus those kindy classes will help you greatly if you ever teach second and third grades in a public school with those kids very short attention spans.