I see that nobody have posted an answer to your question. I'll try to help you with some info on your issue.
First of all, you don't need a F-1 visa while in US. You'd need that if you go back to your country and then try to come back. What you need is to adjust your status to F-1. You can do that by enrolling in a US school as an international student. If you get approved as international student, you'll get an I-20. There is no such deadline imposed that you are speaking of. What you need to keep in mind is that if you go beyond your grace period and then try to apply for an I-20, then you might be in trouble.
You should find a school that you would like to attend and talk to their DSO or an international student adviser. Usually, it does not take that long to approve an I-20 once you get accepted by the school and meet all the criteria for the international student status (F-1).
On the other hand, if you are in your country and trying to get a visa, then you should talk to the US embassy in your country. In my country they issue a US visa the same day.