Would you be renting if you moved?
I've moved my children from the Uk to Australia. They were ages 10 and 7 when we moved. My older DD was/is a very sensitive sould too. I was very worried about it as all her friends in primary would be going off to UK high schools and she'd have to begin a middle school in a completely different culture.
It hasn't been without it's hiccups but by and large 18 months down the line I can clearly see it has been the best choice to make.
I know it's worrying but you can't stand still in life just incase your children can't adjust easily.
You could stay where you are and then in one year's time, he could find he hates his current school or something and then you'd think "Why didn't I move?"
Children have to learn to cope with change in life...it's good for them actually. When he grows up, he will have that experience as part of his armour. Well...not when he grows up but within the next few years!
My only advice is to upsell it too him but don't go on about it too much. Once you make the choice tell him in a matter of fact fashion "We're moving! We're getting a much better house and we'll be near Grandma!"
Don't focus in on "And you will have a new school"
Take each step as it comes. Involve him in the choice of new house...let him see the options so he feels like he has some control.
When he does finally leave his school, try to choose the end of a term.