I am in a slightly similar boat to you. Our 7 year old son may be moving schools again - if so it will be his third. Like you a house move prompted one move. The next move, if it happens, will be due to my concerns that his current school cannot meet his needs.
You probably do this anyway but I think you need to talk to your son about his school day, each day, to get to know of any small issues that are sapping his confidence. Has he made friends? can you invite any of them back for tea? can you offer to help in class or go on school trips to get a greater feel for the children and adults who spend the day with your son?
I am surprised his teacher did not offer you any possible reasons for his anxiety, especially as he must have had a good report from his previous school. Is it worth talking to her again?
Also - and again this may be teaching you to suck eggs, have you looked at the offstead reports for you son's current school?
As for moving him, I think you should investigate the good state school and others. If the school is very good it may be oversubscribed so you will be on the waiting list. We have our son's name on several school waiting lists. We are trying not to lose faith in his current school. I don't want my son to feel I am not committed to it - I'm sure that will demotivate him. However we want to keep our options open. If we are offered a place (there is a possible vacancy coming up at the school we most want him to attend) we will cross that bridge when we come to it and make a final decision. Our problem is that our son likes his classmates and doesn't want to lose his friends. To help him make another move, he belongs to a Beaver troupe held at the other school, so he knows many of his potential future class mates already. If you are unsure about moving your son to the good state school, could you enrol your son in any-out-of school activities attended by pupils at the school?
It's difficult, isn't it? hope this helps!