I think you're right on the mark with the accent. I've spent time in both
New Orleans and Texas and I never heard a native of either place sound
anything like Oswald. Here's an interesting excerpt from Jim Marrs' book
"Crossfire" p.522 (updated 2013 version):
"Gary Mack has reported that three language experts at Southern Methodist
University in Dallas studied tape recordings made of Oswald. They were not
told the identity of the man whose voice they heard. All agreed that the
English words spoken seemed acquired later in life - that English was not
the native tongue of the man on the tape."
I grew up with family members and neighbours who spoke with Eastern
European accents. When listening to recordings of Oswald's speech, I can
hear subtle but consistent traces of this type of accent.