Based on the per-episode crew credits found on IMDB: yes, season two has new writers.
Every episode of season one and two credits Akiva Goldsman, Michael Chabon, Kirsten Beyer, and Alex Kurtzman as the creators of the show.
Season One Writers
During season one, Michael Chabon is the primary writer, with occasional contributions from the other creators as well as a few others.
In addition, Nick Zayas serves as the executive story editor for every episode, and also the writer for episode six.
Season Two Writers
For season two, it appears that the writing duties were mostly handed off to a group of staff writers including Chris Derrick, Juliana James, and Kiley Rossetter. None of these individuals had any credits for season one. Five other writers who were not credited for season one contribute to a handful of episodes, and some of the creators have writing credits on an episode or two as well.
The story editor also changed, with Matt Okumura taking over for Nick Zayas. Nick Zayas is not credited at all in season two.
Here is a table I made, with all the credits for each episode, color-coded by credited role(s):
You can clearly see that an almost entirely different group of people is credited with writing season two.