It is certainly trending that way, and you can see some evidence of conflict within the organization as a result. A few columnists have left the paper, complaining about young woke reporters. There was the dispute about the Tom Cotton “shoot the protesters” op-ed. And, of course, there is the 1619 Project (the ultimate woke icon), which the paper has promoted vigorously and uncritically over the objections of the right and plenty of commentators from the center.
Nevertheless, there is little to fear at this point. There is no evidence of wokeness in the news articles. Liberal columnists still predominate, if by a smaller margin. And the bottom line is that the paper has a readership that is too old, affluent, and white to embrace identity determinism. For the NYT to shift completely to the cultural left would be financial suicide.
If you enjoy the woke perspective, I recommend Vox. The liberal old guard there is gone (many to the NYT), and has been replaced by woke millennials. Reading their material on cultural issues can be an eye-opening, if not altogether pleasant, experience.