I think I ended up going back 3 times, but it is possible to do it in 2 I think, or if you are more organised stay for a few days of you can get the appointments aligned.
The Amphur gave good advice, this is the procedure and you need to make appointments in advance too. This is something I would definitely consider using an agent in Bangkok to do for you. I did not use an agent but regretted it as I wasted time going backwards and forwards to Bkk and you don't have a lot of time on a tourist visa.
we live in the Hua Hin area and my Thai wife's children attend school here. I would say it's the best location for your family and has what you require. Our children love it here and it's completely safe. The airport doesn't go to many places, but Bangkok is only a couple of hours away. Our children go to Sethavidhya school, which is not fully international, but they teach in English and is very nice.
The site is a bit clunky, but I have always been able to get it to work. It is difficult to know what is wrong without seeing your input. Are you typing the exact same information as your landlord did when registering you at the address (do you have their printout to check)?