Attempting to get a long-stay visa while on a 90-day tourist visa

Hello, I came to Berlin two weeks ago with the plan of moving here permanently. I got lucky and already have an apartment and was able to register my address this past week. I'm an American citizen looking to get a freelance work visa and I've been trying to get an Ausländerbehörde appointment for months leading up to being here with no luck.

My question is: if I send my documents and application using the contact form and I don't hear back by the time my 90 days is up or if I get a visa appointment through the website that is a few months from now (as they tend to be) am I allowed to continue to stay in Germany until I hear back? I understand that I would essentially be stuck in Germany until then if that were the case.

Update: Thank you all for your responses, I did manage to get a visa appointment however it's not until October 20th, and my visa-free stay expires July 19th so it looks like I will have to use the contact form since it appears the appointment does not extend the permitted stay.