I think he was a patsy, just as he said. It makes no sense that after the shooting his egress plan was to walk across town to his house, get a 38, and go to the movie theater. It’s pretty obvious he was left out there to swing in the wind. Then he made the statement he was a Patsy, and in jail, tried to call someone in North Carolina. And of course, that number led to a dead end. He was left out in the cold.
https://www.ncpedia.org/raleigh-call-lee-harvey-oswald-and
Maybe he was part of an actual CIA hit team and the plan all along was to let him take the fall. Or maybe they told him he was there on some kind of mission to prevent an assassination. He was a young kid, had intelligence connections, and it’s pretty obvious DC wanted Kennedy gone.
His body language in the video of him saying he was a patsy says it all. He knows he's been set up, thus the use of the termy "patsy." He is angry, but not worried, because he knows he can prove where he was when it happened.
Then they killed him.
We don’t know where Oswald was headed when he was stopped by Officer Tippit. But he panicked then and killed Tippit with his .38. He then headed through a business district where people noticed him and saw him flee into the theater. They directed police into the theater where Oswald attempted to shoot them as well.
Some patsy.
Oswald was a freaking nut case who didn’t think things trough. He operated on impulse like the majority of criminals do.