1. (this is what I do) Use a new, repeat, new blade in a utility knife. Cutting with the grain is almost automatic. Be careful across the grain. Saw with the blade rather that force it through in a single pass or you will crush the edge and cause tear-outs. Don't try to get flush with the knife. Leave 1/32" or so for sanding. Sand the edge down flush applying pressure only on the stroke into the box. Sanding out from the edge will cause tear-outs. To avoid cutting into the previous panel, you might lay a thin metal rule against the bottom of the veneer being cut and cut to this edge. That will give some support and make the cut a shearing action rather than a crushing action.2. Use a down-cut spiral bit in a router. The best bit is 1/2" diameter pattern following bit. I tried a 1/4" bit, which worked OK, but the bearing failed. It was not a ball bearing. I believe that the 1/2" bits do have ball bearings.
Bob