Should I follow up regarding onboarding process?

Hello Developers!!! I am 22M, completed BE in computer Engineering. Upon completion of graduation (May 2023), I worked as an ML engineer (internship); fast forward to December 2023, I had interview scheduled for ML position (off-campus) at reputed company (non-IT), which I passed and after interview in December, I was told to wait for offer letter. Actually, I was referred by the employee of the company. So, till the end of January, I did not receive any update from company regarding offer letter. I asked a person who referred me, he told me that HR has been changed and due to that reason, your onboarding process might be on hold. However, In February new HR got hired and I literally left the hope to receive callback from the company, but after some time I received call from new HR, and again scheduled interview with same manager with whom I was previously interviewed. Manager was quite satisfied with me and asked me If I was ready to join as soon as possible. I assured him to join ASAP. After this interview also I was eagerly waiting for the offer letter; Again, left the hope of receiving any response. However, now it was a beginning of March, and one day I received call from HR, asking for my documents, to release the offer letter. Now it's almost mid-March and again no callback from HR, and no update regarding my joining. It's a journey of 4 months. I never wrote a follow-up mail to HR or manager regarding, onboarding process. Till now I was just asking my referee regarding any updates. It seems cumbersome to him also to ask manager repeatedly regarding my offer-letter. Should I write follow-up mail to either HR or manager. If yes, what should I mention in it? - Thank you