Many places require an air GAP for the DW. If yours has the hose hooked like shown, you have a problem! You need to at LEAST, run the hose in a loop as high as you can get it under the countertop before coming back down to the GD inlet. Also, make sure that you knocked the plug out of the GD on the DW inlet, or it will make a major mess! As it is now, instead of draining to the sewer, when running the GD, you could be pumping waste back into the DW. A high loop will help, but not eliminate that. An air gap will elminate that issue. The lack of an air gap will not affect the sink draining, though.
If the water doesn't have to go up to get to the drain, it should not back up. If the new GD has its outlet lower than the old one, you may need to redo more. A picture would be more revealing than your drawing.