Now, if you look at an interlinear, you’ll see that the entire passage reads in and almost machine-gun style: “and he did A, and he did B, and he did C, and….” In this rat-a-tat-tat, the original contained no verse breaks, no paragraph breaks, no punctuation, no spaces between words, not even any vowels. But there were occasional shifts in the forms of the verbs. If you follow the form shifts to make breaks, you get something like this for 48-51:

And it came to pass that the Philistine arose
 and he was going
 and he was drawing near to meet David
And David was making great haste
 and he was running to the ranks to meet the Philistine
 and he was putting his hand in his pouch
 and he was taking out a stone
And he was mightily hurling it
And thus he was causing the Philistine to be smitten on the forehead
 and the stone was sinking into his forehead
 and he was falling on his face to the earth
 and David was prevailing over the Philistine with a sling and stone
And thus he was causing the Philistine to be smitten
And thus he was causing him to die
And in David’s hand was no sword
 and he was running
 and he was standing over the Philistine
 and he was taking his sword
 and he was drawing her from her scabbard
And he was slaughtering him
 and he was cutting off his head with her
 and the Philistines were seeing that their champion had died
 and they were fleeing

The indented lines represent the places where the text uses a simple imperfect. The non-indented lines are places where some other verb form is used. Now, it seems to me that the indented sections tend to explain or amplify the non-indented lines that precede them. If I am right about this, then all the killing happens after David has prevailed over the Philistine with a sling and a stone (and it all happens with Goliath’s sword).

The Oxford Bible Commentary is of the opinion that two different accounts have been merged (which would also explain this.

The key question is this: When was there no sword in David’s hand?

We could read “there was no sword in David’s hand” as going with the words immediately before in verse 50, in which case there was no sword in David’s hand when he struck down and killed Goliath. In that case, David struck down and killed Goliath without a sword, and so the subsequent statement that he killed Goliath with a sword introduces a contradiction.

Alternatively, we could read “there was no sword in David’s hand” as going with the words after in verse 51, in which case there was no sword in David’s hand before he ran and stood over the Philistine. In that case, we can read “there was no sword in David’s hand, then David ran and stood over the Philistine; he grasped his sword, drew it out of its sheath, and killed him” as an explanation of (rather than an action subsequent to) the killing in verse 50. On this reading, “striking down the Philistine and killing him” means “striking down the Philistine [with a sling and a stone] and killing him [with a sword]“, and there’s no contradiction.

It would be odd for an author to contradict himself in so few words, so absent any grammatical reason for insisting on the former reading I think I’d better give 1 Samuel the benefit of the doubt here.

1. With a sling and stone David prevailed over Goliath.

2. David struck down and killed Goliath.

3. David had no sword.

4. David took Goliath’s sword and killed him with it.

The NRSV compounds 1-3 and makes 4 a separate sentence. This makes it look like David killed Goliath with nothing but a sling (because he had no sword you see).

The ESV renders this in a more neutral way (though they still compound items 1 and 2):

So David prevailed over the Philistine with a sling and with a stone, and struck the Philistine and killed him. There was no sword in the hand of David. Then David ran and stood over the Philistine and took his sword and drew it out of its sheath and killed him and cut off his head with it.

None of this punctuation is in the original. It might have read like this:

So David prevailed over the Philistine with a sling and with a stone. And he struck the Philistine and killed him: there was no sword in the hand of David, but then David ran and stood over the Philistine and took his sword and drew it out of its sheath and killed him and cut off his head with it.