Hi and welcome, Stephen
Your Grandfather's knife was made circa 1944 based on the thong attached via the butt cap hole and the blade grind. It's a "Model 1 All Purpose Fighting Knife".
The sheath is legit! Not one of the usual sheaths supplied by the Shop at the time (Heiser, Southern Saddlery or Moore), but rather Von Lengerke and Antoine. Here is a write up on those sheaths by the guy who authenticates Randalls for the Shop:
http://www.rmkcollector.com/the-rand...andall-sheaths
I don't know where your head's at in terms of sentiment here, but my father bought a Model 3 when he returned from the European Theater in 1945 and gave it to me in 1965. I used it for 3 decades hunting and fishing then retired it. Running across it again maybe 20 years ago, I decided to do some on line research to see if the company was still in business and a whole new world opened up for me. I have quite a few old Randalls and old Randall friends now.
My Dad's knife will be passed to one of my grandsons some day to be further cherished.
Best, Ron