Had to wait a month to get interviewed, then waited another month to be officially hired for a part time job, but then was told I'd probably have to wait an additional month to start training/working. Should I "quit" and look for another job?

Hello everyone! This is my first post and I'm really in need of some advice.
I applied to work at a retail store at my local mall at the very beginning of January of this year, and I waited until February to be called in for an interview. I then was called two weeks later being told I was hired. But I didn't hear back for an entire month so I decided to go in this past week to ask the manager whether everything was okay for me to start working soon. She was nice and told me I just needed to fill out paperwork so she told me to come in on a specific date and fill it out.
I went in to fill out paper work, but it all seemed incredibly rushed and I'm pretty sure the person helping me fill in my login information forgot a step in the process because I haven't been able to log into my work account to put in my schedule. They also gave me a quick 2 minute tour of the place and then at the end just told me to log in the next day to put in what hours I can work which I haven't been able to do.

Lastly, they told me I'd probably have to wait an additional month to get hired (I haven't even started training yet), which I'm guessing is because they're not in need of as many workers right now, but it would have been nice to know upfront because I wouldn't have agreed to work there if I had known how little hours I would be working.

I've decided to give it until the end of the week to see if my login info will finally work before I call/go into the store again, but it's been really frustrating having to wait so long for this job. I'm really tempted to just apply somewhere else that I know will actually schedule me, but I know it's not right to quit before I've even started, does anyone have any advice for what to do in this situation?

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