While Sefaria.org translates this as
But Abram said to the king of Sodom, “I swear to the LORD, God Most High, Creator of heaven and earth:
Rav Hirsch translates this as
And Abram said to the King of Sodom: I have lifted up my hand to Hashem, the most high God the owner of heaven and Earth
Not only is it a matter of having already sworn but the language of הרימותי is specific in that it means that he has already removed any concept of accepting any of the wealth he is now being offered from the world. As Rav Hirsch says:
It is very doubtful what this really means. נשא יד certainly does occur with the meaning of taking an oath but only in God's Mouth. But הרים יד never occurs as swearing except perhaps here.נשא and הרים are different. In נשא it is only the relation to the height to which an object is to be raised which is stressed without any relation to any lower place from which it is to be raised. But this is just what is stressed in הרים. It is not merely a raising bu a "raising out of". Hence the word תרומה. It could therefore here very well mean: "When I went away I dedicated my hand to God, no self-interested motives directed me, but rather I withdrew my hand from all other purposes that are lower than God, and dedicated it solely to God, so that from all my victory I may take nothing for any other purpose, nothing for myself" But the אל ה is against this way of of taking it, in the sense of תרומה it should say לה. Perhaps this "up to God" which expresses was called for to contrast to Sodom, it was to express very definitely not only dedicationg the hand to Hashem, but just thereby the denial of any other selfish use.
With this word he raises his hand away from all the gods up to the Highest, who for him is the Only One. Had he sworn by one of their gods נשאתי would have sufficed.