The shooting of enemy combatants even as they attempted to surrender at Omaha beach did occur, although how many German troops were dispatched of in this fashion is difficult to know accurately. In Beevor's D-Day: The Battle of Normandy he records at least one German survivor claiming that US troops had executed all the survivors of a German bunker, save for a small number of wounded men. In the text, Beevor himself notes there was no confirmation for the above incident but states:

...there were cases of illegal killings, mainly prompted by the violence of repressed fear and a desire for revenge after so many fellow soldiers had been killed.

Certainly many of the German defenders had the expectations that their surrender would not be accepted regardless. One of the machine-gunners at Omaha beach noted to an interviewer after the war:

Also, I knew that we had surely enraged the attackers by killing so many of their comrades. The beach was literally piled up with the corpses that we had shot. So what chance was there of the Americans showing us any mercy now?

There are a couple of reasons it's tricky to know how many soldiers were executed in this manner. For one, these sorts of killings could very often be explained away as simple casualties of the battle. Without witnesses or evidence it is difficult to prove the intentions of enemy troops. Throughout the Western Front, there were many examples of troops on both sides feigning surrender only to immediately re-engage the enemy as soon as troops had lowered their guard. These stories were widely-known at the time and may also have discouraged the taking of prisoners for some men. Waffen-SS soldiers in particular had a well-deserved reputation for executing prisoners and feigning surrender before turning on Allied troops. In turn they themselves were often shot out of hand by Allied troops. Much later during the Ardennes offensive US troops are noted to have taken very few SS soldiers prisoner in retaliation for the infamous Malmedy massacre. Although to my knowledge there weren't any SS troops manning the beach defences, they would come in later as the campaign went more inland.

This does not mean that all of the German defenders were killed. For example the German machine-gunner quoted above was wounded and recounts how he saw American troops that he thought were considering executing him, but who eventually took him prisoner. Substantial numbers of prisoners were captured and present on the beach as follow up waves of troops landed on the now-cleared beach.

Side-note: It's worth noting that in the incident depicted in Saving Private Ryan that OP mentions, the two German soldiers who are shot are both speaking Czech and (from memory) are claiming to have been conscripted and to have not killed anyone. There was a substantial proportion of troops on the Atlantic wall conscripted from Eastern and Central Europe involved in the battle as well.

Sources:

Antony Beevor - D-Day: The Battle for Normandy

Holger Eckhertz - D Day through German eyes