Is there a large drydock and is it free?
Jiangnan most definitely does have a large drydock, intended specifically for carrier construction -- in fact the entire Jiangnan Changxing shipyard is a greenfield shipyard, and during its construction back in the mid-late 2000s there was a highly circulated picture of a model of the shipyard once completed, displaying a carrier within one of its drydocks.
That drydock in question in real life has a length of over 370 meters, and a width of over 80 meters, which is quite similar to the drydock dimensions of Dalian (where 001A is being built and where Liaoning was refitted).
For interest's sake, Huntington Ingalls shipyard carrier drydock (where USS Ford and other CVNs have been built) has a length of about 330 meters and a width of 38 meters -- though that drydock is far more "form fitted" to the shape of a carrier's hull, whereas Dalian and Jiangnan's carrier capable shipdocks are rectangular, larger and able to build a variety of other ships often at once, and are more flexible, so they are intended for different purposes. [though Ford was built at the larger, rectangular drydock at the shipyard of course, which was far more similar in width to the drydocks at Jiangnan and Dalian]
Picture of the model in question
Picture of the real life shipyard, from the last few years, displaying its completion (no carrier of course -- that is yet to come)
But more important than all that, is the constant rumours from credible sources (I think it's been almost five years now) which have consistently stated that 002 will be built at JN.
As for whether it's free or not at this present moment, I am unsure. However, that drydock in the past seems to have only been used for relatively simple, short duration construction of commercial vessels, so things may change quite rapidly within those large shipyards in the space of a few months.
But as I said, when 002 will emerge at JN will depend on a lot of things, including likely the outcome of the competition between steam and EM catapults...