I live in Phuket now. I need my next extension at the end of april 2017. End of may I want to move to Korat.

Should I do my extension in Phuket and later change my address in korat. or ask for more 30 days and make extension in korat with my new address.

There is no 30 more days you can get.

Do your extension in Phuket and then go to Korat immigration to change your address.