"SUNY Oneonta" is the preferred name of the university for most references to it. "State University of New York at Oneonta" is the full name of the university. It is appropriate to use this in situations that establish an obligation against or entitlement of the university, such as a contract, situations in which the audience is likely not familiar with SUNY, such as the advertisement of a vacant position at the university, or situations in which formality calls for reference to the full name of the university, such as some reports within SUNY.
Avoid names other than these. Communication from the university ought not include legacy monikers, such as: "the College at Oneonta," "State College at Oneonta," or "SUCO."