[non-US] German Visa problem

So, my wife and I moved to the greater Munich area a little over two months ago from the US and our status as tourists will be up in about 3 weeks. I have not been able to get a job as an ESL instructor as easily as I had expected. Our issue is that our son was born here just a few weeks ago and he cannot travel. I just need a little more time to get a job offer so that I can begin the application process for our residence visas. Our last finch resort would be to go to Austria for a few months as tourists while we sorted our work and visa status in Germany but our funds are now getting low. This move was a big risk but we strongly felt that it would ultimately be better for our family and for my career. Any advice is greatly appreciated. Thank you. (Please be kind, this is already a terrifying situation).