Many dive shops sell trips with actual dive outfits who then rent regular space on boats. Warm Water Divers do this, so although they were recommeded to me, I found myself diving with several different companies - you pick your dive site and schedule from a table and then WWD phones to reserve - they are basically agents.
Best dive I had was with Calypso Divers off the boat 'Greta' because it had a rule for smoking only on the upper deck and the rule was observed. All other outfits had either no restrictions or they were not enforced and so were a living hell. Worst were Dive Asia (Divemaster ignored stated no smoking in cabin rule) and South Siam Divers (no smoke free areas at all)
Greta is small, a little old, but they serve a good lunch and excellent pancakes

on the return leg.
Only about 50% of shops asked for my certification card and said it was okay if I didn't have it in the shop - it would be checked on the boat (it never was). And don't expect to get a buddy check, even if paired with the dive guide or an instructor. But these would be matters for a qualified diver to worry about - I'd say for OW certification, given my assessment of your written English in your post, go with Calypso Divers as they had instructors with the best spoken English.
http://www.calypsophuket.com/calhell/diving.html
All dive ops are running fine from Phuket though I'm not sure if the Phi Phi dive sites are open yet (again not a big concern if you are doing OW out of Phuket).
Enjoy,
K.