I’ll continue the survey of Galatians to understand the relationship of sin (παράπτωμα) to sin (ἁμαρτία).  Paul hinted at the attitude, activity and the content of the faith of those who were so quickly deserting (μετατίθεσθε, a form of μετατίθημι) the one who called [them] by the grace of Christ and [were] following a different gospel¹ in a series of questions (Galatians 3:1-5 NET).

You foolish Galatians!  Who has cast a spell on you?²  Before your eyes Jesus Christ was vividly portrayed as crucified!³  The only thing I want to learn from you is this: Did you receive the Spirit by doing the works of the law or by believing (πίστεως, a form of πίστις) what you heard?  Are you so foolish?  Although you began with the Spirit, are you now trying to finish by human effort (σαρκὶ, a form of σάρξ)?  Have you suffered so many things for nothing? – if indeed it was for nothing.  Does God then give you the Spirit and work miracles among you by your doing the works of the law or by your believing (πίστεως, a form of πίστις) what you heard?

Here is the same information in a table:

If someone is as enamored with this different gospel (not that there really is another gospel) as I was, a question comes to mind: “Well, what am I supposed to do?”  Paul answered that question like this (Galatians 5:16-18 NET Table):

But I say, live by the Spirit and you will not carry out the desires of the flesh (σαρκὸς, another form of σάρξ).  For the flesh (σάρξ) has desires that are opposed to the Spirit, and the Spirit has desires that are opposed to the flesh (σαρκός, another form of σάρξ), for these are in opposition to each other, so that you cannot do what you want.  But if you are led by the Spirit, you are not under (ὑπό) the law.

Paul then contrasted the capabilities of the flesh (σάρξ) to that of the Holy Spirit.  I’ve put that contrast into a table similar to the one above.

It takes some faithfulness (πίστις) from the Holy Spirit to believe (πίστις) that He will supply all of this, especially in the face of an eruption of sin when the temptation is strongest to take back the reins, as it were.  Paul dealt with that more thoroughly in Romans 6 and 7, but there is some insight here as well (Galatians 2:17-21 NET Table).

But if while seeking to be justified in Christ we ourselves have also been found to be sinners, is Christ then one who encourages sin?  Absolutely not!  But if I build up again those things I once destroyed, I demonstrate that I [old man]⁴ am one who breaks God’s law.  For through the law I [old man] died to the law so that I [new man]⁵ may live to God.  I [old man] have been crucified with Christ, and it is no longer I [old man] who live, but Christ lives in me.  So the life I [new man] now live in the body, I [new man] live because of the faithfulness (πίστει, another form of πίστις) of the Son of God, who loved me [new man] and gave himself for me [new man].  I [new man] do not set aside God’s grace, because if righteousness could come through the law, then Christ died for nothing!

Paul continued this thought later (Galatians 5:2, 3 NET):

Listen! I, Paul, tell you that if you let yourselves be circumcised, Christ will be of no benefit to you at all!  And I testify again to every man who lets himself be circumcised that he is obligated (ὀφειλέτης) to obey (ποιῆσαι, a form of ποιέω) the whole law.

Here regarding the law James mirrored Paul (James 2:10 NET):

For the one who obeys⁶ (τηρήσῃ, a form of τηρέω) the whole law but fails⁷ (πταίσῃ, a form of πταίω) in one point has become guilty of all of it.

So the one who sets out to obey part of the law (you let yourselves be circumcised according to the law) is obligated to obey the whole, and the one who fails in one point has become guilty of all of it.  One who believes these two statements are true searches diligently for another option.  Paul continued (Galatians 5:4-6 NET):

You who are trying to be declared righteous by the law have been alienated from Christ;⁸ you have fallen away from grace!  For through the Spirit, by faith, we wait expectantly for the hope of righteousness.  For in Christ Jesus neither circumcision nor uncircumcision carries any weight – the only thing that matters is faith (πίστις) working (ἐνεργουμένη, a form of ἐνεργέω) through love (ἀγάπης, a form of ἀγάπη).

Both the faith and the love Paul mentioned above are aspects of the fruit of the Spirit, supplied by God to those who believe.  Truth be told the working (ἐνεργουμένη, a form of ἐνεργέω) is from God also: continue working out (κατεργάζεσθε, a form of κατεργάζομαι) your salvation with awe and reverence, for the one bringing forth (ἐνεργῶν, another form of ἐνεργέω) in you both the desire (θέλειν, a form of θέλω) and the effort (ἐνεργεῖν, another form of ἐνεργέω) – for the sake of his good pleasure (εὐδοκίας, a form of εὐδοκία) – is God.⁹  As Jesus said, Do not be afraid, little flock, for your Father is well pleased (εὐδόκησεν, a form of εὐδοκέω) to give you the kingdom.¹⁰

And just in case I think that Paul was a special case, that his crucifixion with Christ was a unique event (Galatians 5:24-6:1 NET Table):

Now those who belong to Christ have crucified the flesh with its passions and desires.  If we live by the Spirit, let us also behave in accordance with the Spirit.  Let us not become conceited, provoking one another, being jealous of one another.  Brothers and sisters, if a person is discovered in some sin (παραπτώματι, a form of παράπτωμα), you who are spiritual restore such a person in a spirit of gentleness.  Pay close attention to yourselves, so that you are not tempted too.

This letter is about deserting the one who called you by the grace of Christ, trying to be declared righteous by the law, being alienated from Christ, falling away from grace.  In context then there is nothing to indicate that Paul shifted gears and began to write about some unspecified παράπτωμα that was of lesser consequence than ἁμαρτία.

Addendum: June 4, 2019
Tables comparing Galatians 3:1; James 2:10 and Galatians 5:4 in the NET and KJV follow.