I can understand the concerns of expats wishing to get some reliable information. I'll never understand however why Thai Visa forums in all the 15 years I've followed them (I don't post much) are so full of conjecture, misinformation, and general BS, much of which causes people to worry unnecessarily even more.
No appointment, no waiting, walk-in vaccinations in the government clinics in the remotest sub-districts with a choice of at least 3 vaccines, and absolutely free. No visas required (the staff don't even know what one is), just a passport or national ID card for filling out the certificate.
I was first quoted ฿2,500.00 per jab at Bangkok Hospital, and to queue for an hour. Why not take a nice drive out of the city to a quiet little town or tambon like I did.
If you don't want a situation like the UK's total chaos - OK, so they are one of the most vaccinated countries in the world, but they have one of the highest mortality rates per capita, but what an administrative mess - get yourselves done today.