Discussions about H1 quotas aside... the issue right now is how to ensure that when you do go to the border, you will be allowed to return.
Since you are out of status, first stop working. Then, if you have been out of status less than 6 months (which seems to be your case), you will be allowed back in, since you will have left US. It is only if you are caught within US, or if you stay more than 6 months illegally that there would be a problem.
You can't file for TN, because you are out of status. Your school can file for H1, but you will have to leave/enter US to activate it. The minimum it will take is a month, since your school has not even filed the LCA for you. So, you face the same border problem whether you try for TN or try for H1, but it will take much longer to get H1 than TN.
My suggestion would be to get TN at border ASAP. The fewer days of illegal presence you amass, the better.
For reference, your legal status is your responsibility.