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