Is Paul making an oath in Galatians 1?

There is no getting around it. Paul was swearing an oath: just as we might do in a court of law, although the wording is less formal than that used in a court. How to judge Paul for that is another matter. The injunction in James 5:12 is the easier one to dismiss, because this is a command attributed to James, not to Jesus. However, the injunction in Matthew 5:37 is more serious, as it is attributed to Jesus.

In defence of Paul, he was writing several decades before Matthew's Gospel is believed to have been published and could not have been aware of what the Gospel would say on this matter. He never met Jesus, and had said a few verses earlier (Galatians 1:15-16) that he never conferred with another person:

Galatians 1:15-16: But when it pleased God, who separated me from my mother's womb, and called me by his grace, To reveal his Son in me, that I might preach him among the heathen; immediately I conferred not with flesh and blood:

A bit late to the party, but I just ran across this question and also an answer to it in Luther's sermons that I thought might be a helpful contribution. The flowing is an excerpt from Luther's commentary on Matthew 5:33-37:

This text has been spun out with many glosses, and many a queer notion and error has been drawn from it, so that many great doctors have been worried about it, and could not become reconciled to the blunt prohibition here that we are to “Swear not at all,” but “let your communication be Yea, yea, and Nay, nay.” So that some have stretched their conscience so tightly, that one doubts whether one ought to take a solemn oath not to avenge himself when he is set free from prison, or whether we are by an oath to make peace and a treaty with the Turks or unbelievers, etc. Now we cannot deny that Christ himself and St. Paul often took an oath; besides, it is said, in the Scriptures, that those are praised who swear by his name; so that also here we must make a distinction, so that we rightly understand the text.

But we have been told sufficiently, that Christ does not wish here to interfere with the secular authority and ordinance, nor to detract at all from the powers that be; but he is preaching here only for the individual Christians, how they are to conduct themselves in their ordinary life.

Therefore we are to regard the swearing as forbidden in exactly the same sense as above the killing and the looking upon or desiring a woman.

Killing is right, and yet it is also wrong; to desire a man or a woman is sin, and it is not sin; but in this way, that we rightly distinguish both, namely, that it is said to you and to me: if you kill, you do wrong; if you look at a woman to desire her, you do wrong. But to a judge he says: If you do not punish and kill, you shall yourself be punished; likewise to a married man or woman: If you do not cleave to your spouse, you do wrong. So both are right, that one is to kill and not to kill, to be and not to be with a woman; namely, that you do not be wrathful or kill, or look lovingly upon a woman, unless you are specially authorized by God’s word or command to do so. If you are wrathful, however, when God commands you, or if you have a wife according to the word of God, then each is right; for what God says and commands is a very different thing from when you do it of your own accord.

As you have understood that, so understand this also; that the prohibition here is, “Swear not at all,” just as he has entirely forbidden killing, so that there may be no wrath in the heart; in like manner, that we shall keep so aloof from man and woman as not to be looking at them, or thinking upon them to desire them. And it would be a dangerous sermon if we were to apply it to the exercise of governmental authority or to married life, and were to say to the judge, Thou shalt not become indignant, or give practical proof of wrath; or to a wedded pair, Thou shalt not look upon or love thy wife or husband: but we must turn about here and teach the opposite, saying: Thou judge shalt be angry and punish; and every one shall have and love his spouse. How then does Christ say one must desire no woman, and have no wrath in his heart? Answer, as said above, he is speaking of the woman that God has not given you, and of the wrath that is not demanded of you, that you are not to have. But if it is demanded of you, then it is no longer yours, but it is God’s wrath, and no longer your desire, but that which is given and ordained by God; for you have God’s word for it that you shall love your spouse and not desire any other. Thus also in regard to swearing; we must see to it, if we have God’s word for it or not.

Consider the phrase "before God" in question in the Greek:

ενωπιον του θεου

This phrase appears in the Greek Translation of the Old Testament in several places.

• Ecclesiastes 5:1-3
• Judges 21:2

1 Guard your steps when you go to the house of God. Go near to listen rather than to offer the sacrifice of fools, who do not know that they do wrong.

2 Do not be quick with your mouth, do not be hasty in your heart to utter anything before God. God is in heaven and you are on earth, so let your words be few. 3 A dream comes when there are many cares, and many words mark the speech of a fool. (Ecclesiastes 5:1-3)

The Ecclesiastes quote is consistent with what Jesus taught. It discourages - but does not proscribe - the taking of oaths. The first issue is pride: promising more than you can deliver and seeking the Holy reputation of one who makes a great promise to God and hopes to not be called out for it when they do not deliver. The second issue is truthfulness. A boastful person who does not fear God may be trying to cover up a lie.

1 The men of Israel had taken an oath at Mizpah: “Not one of us will give his daughter in marriage to a Benjamite.”

2 The people went to Bethel, where they sat before God until evening, raising their voices and weeping bitterly. 3 “Lord, God of Israel,” they cried, “why has this happened to Israel? Why should one tribe be missing from Israel today?” (Judges 21:1-3)

The Judges quote shows a people that are desolate over the civil strife that has killed so many lives. They made an oath before God in tears and sorrow. There is no deception and no pride in their attitude. That is an oath acceptable before God.

Jesus' criticism of prideful oaths should not be conflated with all oaths. There can be a contrite, humble oath, and God will not reject the person who makes and fulfills it. It is important that the person who offers such an oath does so in the fear of the Lord, not in the pride of the heart.