Hi DrDaddy,
if you have all your own kit, including air and weights, then there are options, but don't expect crystal clear water and pristine reef. The coast area around NST is shallow, murky, and is mostly used by fishermen, so expect to find the usual nets, tires, lines. etc.
I sometimes visit a place called Koa Plie Dam, as it has a small rocky area about 80m off the beach. Depth to 7m max and vis pretty poor. Very little to see here, but it's handy to test your kit.
You can sometimes see the change in water colour as you fly into NST, a large brown streak in the water along the coast.
The next problem is getting air refills. Closest place is probably One Stop in Ao Nang (Krabi).
There are many dive operators in the Ao Nang area, but I cannot comment on them or the dive sites they visit as I've never dived from that area. However a few of the dive sites I have dived from Phuket, King Cruiser, Enenome reef, shark point.
Personally, I prefer to drive over to Koh Lanta if I have time for a few days of diving. Scubafish is a very reliable dive center over there.
Sadly, some of these more southern dive sites are being overrun with speedboat trips from Phuket area.
Trang is also another handy place for nice dives, but you'll need your own kit and hire a boat. Alternatively, I know Scubafish do day trips to some of these areas, which are usually quieter.