I think you were like me and arrived during that grey period where they were starting to revert back to Sandbox only. I was supposed to do a day 6 PCR, but never was given a form or any instructions of any sort.
I don't know exact locations, but If you have several days before you travel, just go online and search for Covid testing for travel in Bangkok. You will get plenty of results. For travel, they will probably have to be done in a clinic or hospital, and not at home.
When I arrived on Dec 26th, the first person checked pre arrival PCR test result, Thai pass code, and passport. I assume they now acknowledge that if you have the Thai pass, you have a vaccine certificate, insurance, hotel etc.
Later on, someone requested to see the pre payment for the airport PCR test before they performed the test.
I don't know much about Singapore entry requirements, but if they only need antigen test, then do not worry yourselves with trying to get PCR, and just concentrate on the antigen.
You gave yourself 3 weeks for the pass, so that was good. Some people's literally took minutes. Mine was 7 days. The website says if you haven't received a reply in 7 days, then you can contact them if you are concerned. If there is nothing wrong with the details you entered, a phone call (so I hear) will usually get quick results (if you can get through)