Are there any suitable shipyards in the south?
There are many shipyards but how many are certified for military work? And if they do military work, do you notice they are specialized?
Most notable in the south is Guangzhou Huangpu Shipyard, and its known to produce 056, 054A, coast guard cutter, Type 815G, that SURTASS catamaran I forgot the number, and so on. But the shed they use to make these ships won't fit a 052D. The best bet for this shipyard is to do 054B or 057.
Wuchang Shipyard is in Wuhan, and is known to make 056, including those for export to Bangladesh. They make Yuans. They will likely make the next generation SSK.
Bohai Shipyard does the nuclear submarines. So they will be busy with 095 and 096 stuff.
Guandong Shipyard does replenishment vessels. Maybe they will make more 901s.
Xijiang Shipyard did Type 022 before. Now doing commercial work.
There are also many shipyards in the north near Dalian and around Liaoning province. What you know as DSIC comprises of at least three shipyards.
Because you want to build so many stuff, that kind of crimps on how many you can build on specific stuff.
One of the reasons why I like the idea of building frigates is that you can employ smaller shipyards to pound out these ships. Both Dalian, HDZ and JIangnan also have a lot of commercial ship business. You can also guess that the carrier work with Jiangnan and Dalian also has the potential to distract resources from building destroyers. Maybe other state owned shipyards can be retrained and certified for destroyer work, but there are also questions along the supply chain. How many 346X AESA radars can you make? How many gas turbines can you make? A 055 requires four, while a 052D requires only two which is a reason why the 052D is still attractive. Maybe an 076 requires one, and so does the next generation frigate. At this point, its all managerial.