I've resolved this situation but you may not believe how. From the Properties of the Desktop Icon for the DOS program I went to the Screen tab and chose "Window" (even though I want Full Screen). I then ran the program and it came up in a Window. I right-clicked the top blue bar of the Window to get to Properties and clicked "Full Screen" in the Display Options. In the Apply Properties window that appeared next I chose the option "Save for future windows with the same title." Upon exit the screen went into correct full-screen mode. I then exited the program and went back to Properties for the Desktop Icon and specified Full Screen from the Screen tab. From that point on the program always runs in correct full-creen mode. (Interestingly, if I don't change the Desktop Icon's Screen Properties back to Full Screen, subsequent runs of the program revert to a window.
If anyone at Microsoft is monitoring this board, I ask the simple question as to why this "double clutch" procedure is necessary. Why can't I just set Full Screen in the Screen Property for the Desktop Icon? Also, when Full Screen is selected in the Desktop Icon's properties, why is the Number of Lines box (which says Default) dimmed and therefore unusable?