@Rick *****
When you enter Thailand on May 29, you will receive a new 60-days stay permit which will carry you until July 28. You should exit Thailand until this date.
So, since the visa validity expires on July 29, you need to do a border bounce BEFORE July 28. If you bounce on July 28, you will receive a last 60-days stay permit until September 27. You COULD stay outside Thailand for one night, and re-enter on July 29, that will carry you to September 28
Then it's up to the discretion of the Immigration if they will grant you a 30-days extension on top of that