The original seems to be like this:
1 How sweet the Name of Jesus sounds
in a believer's ear!
It soothes his sorrows, heals his wounds,
and drives away his fear.
2 It makes the wounded spirit whole,
and calms the troubled breast;
'tis manna to the hungry soul,
and to the weary, rest.
3 Dear Name! the Rock on which I build,
my Shield and Hiding Place;
my never failing Treas'ry, fill'd
with boundless stores of grace!
4 By Thee my pray'rs acceptance gain,
altho' with sin defil'd;
Satan accuses me in vain,
and I am own'd a child.
5 Jesus! my Shepherd, Husband, Friend,
my Prophet, Priest, and King:
my Lord, my Life, my Way, my End,
accept the praise I bring.
6 Weak is the effort of my heart,
and cold my warmest thought;
but when I see Thee as Thou art,
I'll praise Thee as I ought.
7 'Till then, I would Thy love proclaim
with ev'ry fleeting breath;
and may the music of Thy name
refresh my soul in death.
In the English Hymnal the hymn is presented as above, except without verse 4. In the New English Hymnal it is presented not only without verse 4, and also with a change to the first line of verse 5 (though with no indication that there has been an alteration). It is that change to verse 5 line 1 that I am interested in.
In verse 5, the poet gives a string of descriptors of Jesus, indicating how he stands to the speaker. The first thought is that he is my shepherd, and the second thought is that he is my husband. What, if anything, is wrong with that second thought? The New English Hymnal has decided that it would be better to excise that thought, and replace it with the idea that he is my 'brother'. (Other hymnals have done this for a long time. 'Brother' is quite common, and others choose 'guardian').
One worry might be that John Newton is a man and cannot have thought of himself as having a husband. But this is surely a very silly thought. There are several reasons for thinking that it is silly. One is that poems are not written to be sung only by their author. This poem is written for anyone to sing, and at least half the content of the pews in his day as well as ours, would be women. It is perfectly possible for a man to write a hymn that is designed to make sense for women to sing. Of course many hymns are not designed to make sense for women to sing, and some use language that tends to assume that we are all brothers and so on. I don't tend to mind that language, but it would be odd to object to a hymn that managed to avoid that, and wrote in a way that is OK for women to sing.
A second reason for thinking that it is silly is this. If the poet wants to indicate that I stand in a spousal relation to Jesus, then the correct term for describing who Jesus is needs to be either "my wife" or "my husband". If, as John Newton quite reasonably assumes, the Jesus we know is presented as of masculine gender, then the correct term would seem to be "husband", not "wife". So finding oneself wanting to say that Jesus is the beloved other to whom we are spiritually married, one should indeed say that he is "my husband", and we should say that even if the speaker is a man. One would only find this odd if one has a fixed idea to the effect that marriage—even spiritual marriage— is, and could only ever be, between a man and a woman, and a man could never speak of his "husband". John Newton shows us that this was not so in 1779, and it is certainly not so in 2018.
A third reason for thinking that it is silly is that it's clear that Newton was not doing this by accident. Indeed the first verse speaks of the believer as "he", so he's not pretending to be a she. Nor is he writing from the point of view of the church or of the soul (both potentially female images who can be conceived of as the bride of Christ). The verse speaks in the first person (accept the praise I bring), and has mentioned the spirit, the soul and the heart in other verses as other parts of oneself. So it is not the soul speaking.
Of course, the image is striking and makes you sit up and think. But is it that the hymnal editors think that it is inappropriate? Or they (male editors) can't stomach encountering language they wouldn't themselves use? Just what exactly is their problem, and why do we even find the image a surprise? We're happy with the thought that Jesus loves us, and that we are his children, his brethren, or his friends (none of which is literally true in the normal sense of those words). Why exactly would we baulk at the idea that he is also our husband?
One option is that marriage and spousal relationships seem to be one-to-one, and therefore claiming Jesus as my husband might seem to exclude you and others from having the same relationship. But it has always been a normal part of entering a convent for the novice to be imagined entering into a marriage with Christ, and if we can cope with that idea for the monastic life, the objection that Christ could have only one spouse is no longer a sound one.
I suggest that we leave Newton's poem as he wrote it and stop trying to enforce a false image of what the church means by marriage, and the false dogma that, for the traditional church, the term "marriage" can only be used of a relationship between one man and one woman.