As I understand, the employer has to be the petitioner- which means they have to file it on behalf of you. The department has to sign the I-140. I am not sure about the petition letter. An employment letter from the department alone will not qualify. Consider this as something like your H1, where the employer files the petition for you.I do have a support letter from my Department head. He express that he strognly endorse my application and claim that I am important componet of the dept. But the letter is actually a reference letter. Is that enough?
Does the Dept. has to sign the I-140 and the petition letter?
No, you can't. I tried. The University MUST be the petitioner. You can certainly do it yourself, but the University must in the end sign the petition. You can get a lawyer, pay for the lawyer, and the University can agree to be represented by the lawyer, but in the end the University must petition for you, and they must sign the petition. The Executive Vice Chancellor's office is usually in charge for these approvals. Its easy if you are tenure-track faculty, but often difficult if you are not.
If you already have the option of getting on H-1b status now, why wait until you get a rejection of your I-140? Get on H status ASAP.... I assume you have your final USCIS J-1 waiver, if applicable.
I know you want to work another job - but the bottom line is that you should evaluate what is more important - staying in your current job and getting on H status now, or using your EAD for 2 jobs and risk not being able to "jump" to H status in time. Remember you can't "jump", you have to apply and wait the standard processing times etc etc. How strong is your EB-1 case? Only you can decide the risks you want to take...