The following was supplied by this site which makes use of services provided by the United States Naval Observatory. Actual times may vary slightly based on where in the US you were.
To answer your questions:
It was a Thursday, not a Saturday.
It was not yet a full moon but was in the first quarter.
Sun and Moon Data for One Day
The following information is provided for Dallas, Texas (longitude W96.8, latitude N32.8):
Thursday
13 September 1956 Central Standard Time
SUN
Begin civil twilight 5:45 a.m.
Sunrise 6:10 a.m.
Sun transit 12:23 p.m.
Sunset 6:36 p.m.
End civil twilight 7:01 p.m.
MOON
Moonrise 1:54 p.m. on preceding day
Moonset 12:20 a.m.
Moonrise 2:38 p.m.
Moon transit 7:54 p.m.
Moonset 1:13 a.m. on following day
Phase of the Moon on 13 September: waxing gibbous with 66% of the Moon’s visible disk illuminated.
First quarter Moon on 11 September 1956 at 6:13 p.m. Central Standard Time.