Yes, there were enough German reserves that without continuous coverage, the Germans could have split the Allied lines and started flanking the weaker half of the invasion and pushed it back into the sea.
There were considerable German reserves further into France.
What saved us was our ability to command the skies and send in fighters to strafe and attack trains, truck convoys, troop columns at will. Not much is written about the number of fighter sorties launched in support of Normandy, but it was considerable effort on the part of the fighters and medium bombers to eliminate German reserves further inland.
The other thing that helped at Omaha was the decision to abandon the plan to open the beach exits from the front. Rather, people went over the hill, then came around the back against the fortifications.
“Saving Private Ryan” gives a good flavor of that effort. “The Longest Day” doesn’t.