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Regarding the issue of carrier, it does say clearly on the website when you look at a flight listing "Operated by United Express," before you booked the flight. Not sure that Cheapoair is responsible for the fact that you missed this.
Regarding the refund, if you have proof they said that you could cancel your tickets and get a full refund (less some service fee), and they refuse to abide by that promise, then do a credit card chargeback.