As Chris said, you need to wait 2 weeks after your second shot to be seen to be fully vaccinated and you'll need proof of it for entry, not necessarily for the COE.
You can enter Phuket sandbox for 2 weeks then carry on to elsewhere. Alternatively you can do an asq for two weeks in Bangkok. Fully vaccinated were only going to have to do 1 week quarantine but the delta varient changed that so it's 2 weeks regardless