Are the Chinese taking market share of making these LNG carriers from the Koreans?
Making LNG carriers requires some of the most sophisticated technology, probably more so than making DDGs.
Yes and yes. LNG carriers can be considered the top end or crown jewel of ship building. High tech ship building is one of the ten pillars of Made in China 2025. You need temperature and pressure sensors throughout the length of the ship, monitoring for any leaks and changes. As some of the LNG goes back to gas, you alleviate the pressure by using the gas as fuel for the engines, so the ship uses its own cargo as fuel. You can expect these ships to be vital to a nation's energy security --- there was political ruckus in the US, realizing that the US completely lacks the ships to secure its energy security, that its natural gas industry is completely dependent on foreign vessels. If you don't have enough vessels to transport the gas, transport rates goes high, and exporting gas from the US becomes uncompetitive versus Russian pipelined gas.
As late as ten years ago, Japan was dominating this particular segment of the ship industry, then some years later the Koreans pulled the rug out and under the Japanese ship building industry. Now they dominate, until Chinese shipbuilders start to chip against it. This is why LNG carrier shipbuilding is a big feather to the major Chinese shipbuilding yards, including Dalian, Jiangnan and Hudong Zhonghua, and you can argue, might be more important for their respective companies over the military builds. Building these ships are both a big part of the Made in China 2025 and for China's energy security. These ships have to conform to all the safety standards of the countries the ship visit, and this can mean multiple countries, and in addition to that, even some kind of protection against terrorist attack --- you don't want a ship like this going boom in your port.
Sorry if this is out of topic. Just saying don't be surprised to see more of these ships being produced in the best shipyards in China alongside the warships.