Is the front of your board slightly cupped? This gap can be caused if the box front isn't perfectly flat. The clamp on the D4R will hold it flat while cutting, but the cup will return when the clamp is released. If there is any play at all in the dovetail and the board cups, you will see this gap as the face board pulls away at that point from the side board. Good clamping should be enough to pull this small gap closed. The glue should keep it from opening back up.
The 1/16" offset is usually caused by a slight difference in side position of one piece being cut to the other. When doing 1/2 blind dovetailing you are using different clamp positions for the two boards. For through dovetails, you are using only the front clamp position, so both boards would be referenced against the same side stop. I'm suspecting that there may be a 1/16 difference between the two side stops (top and front) in your jig, but since the D4R base is CNC machined, I'm not sure how this could happen, but it's still a possibility.
Can you place two boards, one in the top clamp and one in the front clamp with both up against their side stops and then check to see that the edge touching the stops of both boards are in line with each other (the edge of one directly above and in line with the edge of the other)? If there is a 1/16 difference in the stops, there is a machining problem with the end stops on your jig. If not, we need to look further.
Another possibility is to watch closely to assure that the boards do not move away from the stop as you tighten the clamp, although this happening by the same amount each time is very unlikely. but we still need to check this
Central North Carolina