Not an IT expert, but could it be that user name and password are saved in your browser and automaticly filled in. Try to clear those savings and try again.
If you want to stay long term with your child in Thailand, at the age of 6 the child needs to go to school. Contact a school and apply with documents for the Non Immigrant ED visa for the child.
Every foreigner needs a visa, that there is no fine for children until 15 years of age, does not change the rule.
Problem? The regulations of the Thai Immigration if you like to transfer a 1 year Extension of Stay into a new passport are set by Immigration.
You get from your Embassy a letter that the new passport is a replacement of the old passport with the passport numbers, you copy old passport and ID page new passport and Immigration will than put a new stamp in your new passport.
Always good that people ask to be sure, but every 90 days you went with the school to Immigration and signed the STM2 form, see example.
Show this to the school and maybe they will ask Immigration to understand what it means to inform Immigration when the reason for the stamp change. a Change is ending education in your case.
FIRST thing to do is visit your local Immigration Office, this as NOT every of the ove 80 offices are certified to perform the conversion from Visa Exempt Entry into the Non Immigrant O visa,
Better be good prepared than finding out you run out of time to arrange all things at the last moment.