Okanas

Again, no employer "runs" I-9s or does anything with them besides storing them unless and until DHS performs an audit of that employer.
 
For a state governmental agency, that's going to be unlikely.

Those I-9s rarely reach DHS. They are just usually held by the company and then if they are audited by DHS they then have to turn them over. 
 
If anything ever happens you can explain it at that time, but it's highly, highly unlikely. 

Congratulations on the new job! I recently switch companies too and I only presented by "clean" SSN and Driver's License as for the I9 I think I just mark as LPR. I had all my extensions letters, I-551 stamp and expired green card with me in case my employer asked for them but the SSN and DL was enough.

On an I-9 you would check the box for lawful permanent resident.
 
As you say, you can use your valid DL and social security card as your I-9 verification documents, and don't need to use your green card if you don't want to.