Actually, Oswald did drive, and I believe he owned a car. He drove out to where he took the pot-shot at the retired military officer ( General?) who was head off local John Birch group. Immediately after the shooting, when numerous witnesses said they saw the shooter in the TSBD, police went in, Oswald was the only person unaccounted for. As police began the search of the building, they encountered Oswald leaving, An employee told police he ( Oswald) worked there, and they let him go. He left via bus, then when on foot was contacted by Officer Tibbets, since Oswald matched the description of the shooter given by eyewitnesses. Oswald shot and killed Tibbets in view of numerous witnesses, fled to movie , snuck in, and the rest is well know.
An interesting point, oftentalked about by conspiracy buffs, is the bullets recovered from Officer Tibbets body had no ( or very little) rifling marks, so they claim it was an "assassins special": actually is was a milsurp S&W originally chambered for .38 S&W ( NOT special ). This has a larger bore then "special". To make the revolver more attractive to buyers, the importer cut the chambers longer to take the .38 SPECIAL round, but the bullets for the special, being .357, were way undersize in both the chambers, and the bore, hence little or no rifling marks. Typical of Oswald, it was all he could afford, but it worked.
Read the books I mentioned, keep an open mind, and report back.