You can change brokers and have all of your shares transferred. I have done this before. I was actually selling my shares through Fidelity--not an endorsement as I have only used Robinhood, Fidelity, and Vanguard. I don't know who the best broker would be.
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I've been thinking about e-trade. I'm going to have to do some research. I really did love the simplicity of the Robinhood app though. I've only got a few positions left open which should make the process much simpler.