First, we know that BA can violate the three AS Oaths with impunity:
However, the BA still have the Ageless look and the shortened lifespan which are characteristic of people bound by the OR. They have to, in order to blend in with non-black AS. So, it is likely that the "new trinity" of Dark Oaths sworn by the BA are taken on the Oath Rod. There is further evidence for this. Galina knows that Oath Rods can be used to remove Oaths: "If [Sevanna's "binder"] was a second Oath Rod, it could be used to remove any oath she swore now" [TPOD: 11, Questions and an Oath, 255]. This is not general knowledge among AS (Seaine and Pevara, both high-ranking AS, had to figure that out by themselves), so we can conclude that she knows it from experience. That is, the BA are freed from the Three Oaths via the Oath Rod. It means that the BA induction involves the Tower's OR-- Galina is only familiar with the Tower's Rod (Number 3). It's therefore reasonable to guess that the Black Ajah Oaths are administered via Oath Rod.
According to Verin, one of the Oaths is "I swear not to betray the Great Lord, to keep my secrets until the hour of my death." [TGS 39: A Visit From Verin Sedai] This explains why the golden girls can't get Ispan to spill her guts about the BA no matter what they do [TPOD: 20, Into Andor, 399].
Also note Joiya Byar's "confession" about the supposed BA plan to use Taim as a False Dragon to discredit Rand. That obviously never happened, and so it appears that Joiya lied, while her fellow prisoner, Amico, told the truth about the plan to get something in Tanchico to harm Rand. This fits well with the idea that the BA swear an Oath not to betray the Ajah. Amico, having been stilled, was released from any Oaths that she took on an Oath Rod. Thus, she could spill the beans when pressed. Joiya, however, had no such release, so had no alternative than to either lie, or to grovel saying "I'd LIKE to tell you, really, but I can't!!" [Laura Parkinson]
For the rest, we can make reasonable guesses. There is almost certainly some kind of oath of servitude to the Great Lord, the same kind of oath the Forsaken have spoken of swearing. There is also likely an Oath of secrecy.
However, the Oath is probably more specific than "keep the Black Ajah a secret", since BA have told outsiders that the Black Ajah exists - Liandrin for instance [TGH: 5, The Shadow in Shienar, 66]. Adeleas said that in particular she could not force Ispan to reveal the names of other Black sisters no matter what she did. So the second Oath is likely along the lines of "do not reveal the identity of any Black sister to someone not sworn to the Dark Lord", or some such [David Chapman].