welcome to Wefugees and thanks for reaching out to us with your question.
First of all, I'm sorry to hear you can't reach the immigration office. From my own experience, the authorities are generally difficult to reach. Have you tried to contact them by e-mail? Here is the list of e-mail addresses you can contact: https://www.stuttgart.de/vv/leistungen/auslaenderbehoerde-terminvereinbarung.php.
The best way is to write only one e-mail to the responsible team (it results from the first letter of your surname). If you have already sent an e-mail, I can only recommend you to be patient. Due to the high demand for appointments, also the processing of the mails unfortunately takes longer. In any case, I keep my fingers crossed that you will get an appointment soon and wish you all the best.
Please, don't hesitate to get back to us again with any further questions.