PD sends to state, state sends back to PD. Just did my renewal early and it took 6 weeks from application to getting it in the mail.
I am in a small town, and the clerk at my local PD that handles them is a personal friend, and I also asked her to fast track it on her end, because of the new bullshit incoming.
I do think a lot of these get hung up on PD desks.......my clerk told me the state generally takes about 4 weeks or so at least. So if your getting jerked around for months, its likely your local PD.
Other than that legally having to get a permission slip to own a firearm....you should feel nothing but sadness.