Based on the latest wiki entry (dated Sept. 20, after Week 2), the decision was still up in the air. Head coach Todd Monken stood by Watson for Week 2 despite his rough Week 1 outing (205 yards, 1 INT, 5 sacks), but the team explicitly stated they would revisit the starter decision before Week 3 at home vs. Carolina (from the Deshaun Watson page).
There isn't a final verdict recorded in the wiki yet — it captures the situation as "under evaluation." Shedeur Sanders has been waiting in the wings, and Dillon Gabriel (who had a strong rookie season with 7 TDs vs. just 2 INTs) is also in the QB room, so it was a legitimate shakeup scenario. You'd want to check the latest Browns beat reporting for the final call on who took the field against Carolina.