Should I call places back I've applied to?

Yes.
“Hello, this is @pleiades. I submitted my resume’ on July 1 and I was just calling to follow up. Has this position been filled yet?” If they say no, you can say something like “do you know when they will be interviewing?”
There is no reason not to do this. It shows you are really interested.

Yes.

It may not help, but it won’t hurt. It worked for my brother with his first job years ago.

Often, though, I’ve heard it’s hard to get anyone who isn’t too busy to listen. You may get a harried floor employee who says, “They’ll hire one of thee days. They’ll call if they need you.” If it’s a small business and at specific job, you might be able to get attention by asking a more specific question about it. (“And you’re expecting someone who can’t work during school hours, right?”, “That doesn’t start until after Labor Day, correct?”). Just do your best to leave an impression that makes someone curious about your application.

It may help to call even if the position is filled because it shows that you’re trying harder than everyone else that didn’t call and they may remember your name for a future opening. It’s better than waiting in the dark wondering if they read your application or not yet. Do your best to get a date for when you expect to hear back from them.

Thanks for the encouragement and advice!

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