James is a practical Epistle. Sometimes under the guise of general principles and theology a person can evade the forcefulness of the scripture and James in some ways avoids general principles (as dangerous as that could be at times) to pin down the hypocrite, who does not live according to them though may be very conversant about them. As such is the case, to understand the theology and principles that present the context of James, ironically we really have to turn to other Epistles.
For example a more doctrinally positioned call to ‘draw near to God’ can be found here:
19 Therefore, brothers and sisters, since we have confidence to enter the Most Holy Place by the blood of Jesus, 20 by a new and living way opened for us through the curtain, that is, his body, 21 and since we have a great priest over the house of God, 22 let us draw near to God with a sincere heart and with the full assurance that faith brings, having our hearts sprinkled to cleanse us from a guilty conscience and having our bodies washed with pure water. 23 Let us hold unswervingly to the hope we profess, for he who promised is faithful. (NIV, Heb 10:19–23)
When comparing this to James one obvious thing seems to be worth noting. James is using language more appropriate to sinners rather then believers. James says ‘You adulterous people’, ‘you sinners’, ‘enemy of God’, etc. As scripture does not use this type of language with respect to genuine believers we must understand that the direction is for ‘sinners’ to draw near to God, whereas the parallel Hebrew reference is assuming faith in those verses.
It is a common fallacy in biblical exposition to assume that believers are always the intended audience to every verse in every Epistle but that is sheer nonsense. One must always use the context and choice of words to determine if the verse is directed to believers, unbelievers or those who might be one or the other based on their unclear commitments and obscure confessions.
What does it mean for a double minded man to purify his hearty and wash his hands? It means that by nature man is away from God and must humble himself to draw near to his throne. ‘God opposes the proud, but gives grace to the humble’. In the context this must mean that a sinner must recognize he is one and needing salvation or face the eternal wrath of God. In repenting and believing in Christ he must turn from the practices of sin (i.e. cleans the hands that are otherwise engaged in acts of evil). However the gospel does not ascribe any real righteousness to a person who only repents outwardly without faith. Therefore the heart must also be cleansed by receiving Christ through faith, making a soul righteous and justified before God.
By cleansing the hands and the heart through faith a person will no longer be double-minded but will become a Christian. A double-minded man is not a description of a Christian struggling with sin (as in Romans 6), it is a description of a wicked hypocrite:
A double-minded man, δίψυχος ἀνὴρ] The word signifieth one that hath two souls; and so it may imply, 1st, A hypocrite, as the same word is used to that purpose, “Purify your hearts, ye double-minded, δίψυχοι” (Jam. 4:8). As he speaketh to open sinners to cleanse their hands, so to close hypocrites (whom he there calleth double-minded, as pretending one thing and meaning another) to purify their hearts; that is, to grow more inwardly sincere; and so it suiteth very well with that phrase by which the Hebrews express a deceiver, “With a double heart do they speak” (Psa. 12:2): in the original, with a heart and a heart, which is their manner of expression when they would express a thing that is double or deceitful; as divers or deceitful weights, is a weight and a weight in the original (Prov. 20:23). As Theophrastus saith of the partridges of Paphlagonia, that they had two hearts; so every hypocrite hath two hearts, or two souls. (Manton, T. (1840). A Practical Commentary, or An Exposition with Notes on the Epistle of James ,p. 39)
Respecting a drawing near to God within the context of a believer the story is different. A believer has been made perfect before God through an external righteousness charged in a court against him. This has effectively opened up a living way as a result of God drawing near to us in Messiah. Therefore rather then recognizing that we might be ‘enemies’ and that God might be ‘opposing us’ and that we should therefore ‘mourn’ and be ‘humbled’, we have already been humbled and already do mourn, therefore the key words in our approach are to be with boldness and a ‘full assurance of faith’. We are to boldly approach the throne knowing we are accepted and 100% righteous through grace. The contrast between a ‘sinners’ approach to a God of judgment and a ‘child’s’ approach to Father that has adopted him, is very different in deed.