My opinion…do a couple of things.
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Keep reminding yourself why you chose this smaller college. Continue to do your best there, and try to get involved in some club or something that interests you.
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Send a transfer application to your public university. And maybe look at other options. But you seem to like the options available at that public university. So…apply to transfer starting at the start of your second year.
This way…both options are open to you (assuming you get accepted again) and you can make up your mind in the spring.