...but kept the project public since we are not willing to pay the fee to host private projects on Github
This is tangential to the rest of the answer, but there are numerous services that allow free hosting for private git repositories, including Bitbucket and GitLab. Github is great for project visibility and accessibility, because so many people are on it already, but if your goal is to reduce accessibility (i.e., privacy) then that factor has minimal (perhaps even negative) weight.
However, we are concerned that someone could steal our code before we public it to the Apple App Store. We are worried that someone publishes or submits our app as their own to the app store before we do. How can we make sure that this is not allowed?
If haven't licensed your code to anyone, then no one can legally republish it. If someone does illegally republish it, then you can issue a DMCA takedown notice to the App Store.
What kind of license would allow an open-sourced project to be protected in a way that the owners maintain those publishing permissions?
Such a license could not satisfy the OSI's Open Source Definition. Under an open source license, everyone must have rights to republish the work:
3. Derived Works
The license must allow modifications and derived works, and must allow them to be distributed under the same terms as the license of the original software.
Rationale: The mere ability to read source isn't enough to support independent peer review and rapid evolutionary selection. For rapid evolution to happen, people need to be able to experiment with and redistribute modifications.
If you are interested in restricting how downstream recipients can republish your work, then you are not interested in licensing your software at open source as the OSI (and FLOSS community at large) defines the term.
What kind of suggestions do you have for open-sourced projects that still want to maintain ownership of their projects?
If you choose to use an open source license, you may exercise trademark control over your project's name, so that no one else can republish it with the same (or a confusingly similar) product name. Your project is yours and trademark gives the legally ability to have your unique work recognized as yours, but if you license your software under an open source license, there is, by design, nothing to stop someone else from forking your software and calling it by a different name.