Continuing with a postdoc with my PhD advisor, but have not got any official offer letter. Should I be worried? I am an international student who's currently in a bureaucratic mess regarding my work permit application. I will get my permit possibly in February end. I completed my PhD thesis submission in December. Now, my PhD supervisor offered me a postdoc position soon after my defense, but I cannot officially sign a contract till I get my work permit. So, I don't have any official offer letter with me at the moment.

Should I be worried that I currently have no letter indicating the appointment and how can I handle the situation better?
No, you should not be worried. It is perfectly normal for official offer letters to be delayed.  Ask your supervisor if they submitted the necessary paperwork.  Ask HR what needs to be done.