For those whose only problem with a name change is the publishing requirement, just rent a cheap apartment in an adjoining state, or in a state where it's easy and cheap to fly to and back in a few hours. Visit every few weeks until residency requirements are met. (I've heard some states have no time requirements, just have an apartment and utility bills in your name. I haven't checked that claim out for myself.) Then publish in a newspaper in that state, change your name to something very common, get a driver's license in that state, change the name on your SS card.
As soon as your SS card is changed, get a DL in the state you won in and claim the JP. If you use the DL in the adjoining state, that would make it awfully easy for someone to find your name change when the lottery releases that John Doe from Name Change State claimed the JP. So change it back to your state in case someone becomes suspicious they won't have any idea where to start looking.
Then go to yet another state and change your name back, or to another name you want to be known by for the rest of your life.
My state doesn't require a photo or a media event, but they do post on their webpage that there's a $$xxx winner in the lobby, alerting media or whomever else is interested to mob the place. So you'd definitely need to wear a disguise to get in and out without revealing your identity. Definitely don't take your own car. It's not that hard to trace a license plate.
Yes, this is a whole lot of trouble, and takes extra time before you get your dough, but for some of us it's worth it. Some of us have lunatic relatives who'd be happy to put us in the ground for a few thousand dollars let alone a few million. Some of us have enemies that are very resentful and wish the worst upon us who would finally be shoved over the edge upon learning we are wealthy and they're not. Some of us have acquaintances who would lie and say half the ticket is theirs and keep it tied up in court until they got something for their trouble.
To answer the OP question, if I couldn't stay anonymous with a trust, then I'd definitely change my name in order to do it.