Applied at city for Unknown

I interviewed for a city job about three weeks ago and was told to expect a decision in a week and a half. After a week, my references were contacted, and I followed up at the two-week mark to check on the process. The recruiter mentioned to give them one more week. Now, that extra week has passed, and I still haven't heard anything. I'm considering sending a brief follow-up tomorrow if I don't receive any updates today. I know that government processes can be slow, but I'm also trying to manage another job offer that I signed. This city role is significantly better than the other position, and with my start date for the other job in a few weeks, I'm really hoping to get some clarity soon.