I should also mention that I'm already IN state gov't, and the HR person knows me fairly well.

Do you guys think I should mention the application I've already sent in for the first job in my cover letter for the second job?
Yes you should.

No, you shouldn't. If they notice it, it shouldn't be a big deal, and it they don't, who cares?

If the state gov't is like larger corporations I've worked for, many positions require interviewing several candidates. So in a way you may be doing them a favor by jumping into two opportunities when you can only accept one.

I'm not sure why it is okay for employers routinely interview several candidates for positions, but candidates get all weirded out by the notion of having multiple irons in the fire.

The worst thing that will happen is that you will prefer one job over the other but will be offered the other and have to decide whether to take the one that's offered while the other one is still in play.

If the HR person knows you, maybe sticking a handwritten note in with the second application, something like "Seems you have a lot of great jobs opening up! Hope to talk to you about some of them!" or something along those lines? Or if you hand-submit the application, just saying that? (Or do you not know her quite well enough for that?)

I wouldn't think you needed to mention it formally, but it would feel weird to me not to acknowledge to someone who knows me that I was handing them multiple things.

If that makes any sense. Which it probaly does not, because I'm posting pre-coffee again.

That makes sense to me, occhiblu, and it seems the best option.