What you have done is request they re-open your claim. you have a "quit issue" on your part time job. So what they will do is make a decision about whether you quit the part time job with good cause. In the meantime, since you still have a claim open, keep filing your certifications for each week that passes where you are not working anywhere. If your claim is approved again, they will back pay you for these weeks you have missed.
Yes, you did right, you had to tell them about quitting the part time job. That was the only honest way to handle it,but now they will stop your claim until they can get a decision about whether you had a good reason to quit.
Moral of the story is, do not ever quit anything until you have the hire letter signed for the new job. This happens all the time. But maybe you will be re-approved. The decision to grant benefits on the job you lost in May was what got you going in the first place, but you had this other part time thing, and you did voluntarily quit it, so it now becomes your separating employer for unemployment purposes. The decisions will have to be made from now on based on the part time job you quit.
But be sure you emphasize how sure you were that you were going to get this new job...and in the meantime, be looking very hard for another job, because they may not decide in your favor. Appeal, if you are denied, you have nothing to lose. But you did quit, a bit prematurely it appears, though it seemed to be a good reason at the time.