Zealousideal-Egg1354's journey

I recently interviewed for a Data Scientist position at an insurance company in the US. After my first interview, they decided to consider me for a more senior role as a Senior Data Scientist. However, after the final round, which included a video call with HR eight weeks ago, I was informed that they wanted to hire me for the Data Scientist role in a different team. HR mentioned they needed headcount approval, which could take 2-4 weeks or even longer as they assessed the needs of other teams. I was caught off guard by this process and expressed that I was in the final stages with two other companies. I made it clear that I might not wait long if I received an offer elsewhere. Since that conversation, I haven't heard back from HR, which has been frustrating. I did reach out to the hiring manager two weeks ago for an update, but I received no response. I have a tight deadline to secure a job offer by July 6th and start by July 12th due to my visa situation. Chasing after companies feels uncomfortable, and I hate the idea of appearing desperate for a job. I'm seeking advice on how to navigate this situation, especially since waiting for this company feels like I'm not their priority.