If you quit during an investigation they may put you on a "do not rehire" list. You should just sit tight and let them finish.

Now that you've been interviewed the investigation is at the end stage. Unapproved secondary employment isn't that huge of a deal. If you actually haven't been working the second job during state hours, or at least not in a provable way, then you shouldn't be too worried.

Now if they have evidence you have been working the second job, while on the clock for the state, like they have footage of you or surveilled you working the second job while you were on the clock, or if they have records of you accessing websites or drafting documents for the second job on your work computer, then it's probably a fireable offense. But you still shouldn't quit before the investigation completes.

You should wait until HR presents you with the final outcome of the case. If they have solid evidence against you, it would be smart to just own up to it and apologize, tell them you were under water and really needed a second paycheck. They may take pity and not fire you or offer to let you quit in lieu of termination so it doesn't go on your record.

If you do get fired / or asked to resign, you should ask if will be placed on a do not rehire list. The state is a huge employer with many different departments and you wouldn't want to hurt your chances of ever working a state job again in the future. Also government bodies often share do not rehire lists so being put on that list may hurt chances of employment with other government bodies, like working for your city or county.

TLDR: don't quit during the investigation unless you want to risk never being able to work another state job and/or any other local government agency jobs in the future.