It sounds like you have an Impala Custom.
The Molding under the emblem should have studs and nuts on the back side that you need to get from inside the car.
On the Rear Drip Rail, if you ever find out how to remove it, let me know, lol.
I have taken off all kinds of Moldings through the years, but that Molding is the only one that ever stumped me. I went to the point of almost damaging it, when I finally gave up.
-Parting Out over 75 '58 to '73 Full Size Chevy Cars-
-National Impala Association
-Vintage Chevrolet Club of America