Corporations have to file with the state and disclose their officers and boards. Trusts, afaik don't have those obligations. At least I don't believe that they do since the one time that I tried to get information about a trust that held a piece of property, I got absolutely nowhere.
In terms of buying the property, the simplest way is to buy the property outright and then transfer it to the trust. That way anyone looking at public records won't see your name associated with the property. They would have to look up the title history. Unfortunately in many places like my state, that information is also online.
So my guess is that you would have to transfer assets sufficient to pay for the property into the trust and then have the trust make the purchase. That should work. But that's just a guess.
Aside from that, I can't think of any way that you wouldn't leave some sort of paper trail that wouldn't be a matter of public record. Maybe if the trust had a verifiable source of income that could qualify for a mortgage, the trust itself could get the mortgage, but again, that's just a guess.