I found a lovely article by Bill Dagle on the origin of this hymn and will borrow a bit from him so that you can have a bit of background.
One of Fanny's dear friends was William Doane, a factory president and active Baptist layman.
It was while Fanny was visiting the Doane home in Cincinnati that she was inspired to write the hymn "Draw Me Nearer." It was a late afternoon, and the family began describing to their blind friend the beauty of the sunset. They talked about the wonder of God's creation and the desire to live a life pleasing to Him.
Fanny asked Doane to write down the words to a poem that was forming in her mind and in just a short time the words to "Draw Me Nearer" were completed.
"Many will go through this life never seeing the things that blind Fanny did. In her blindness, she saw a loving Father and a caring Son who opened the door to heaven for those who would accept and believe. She also saw a day when, on the other side, all things would be made new, including her own eyes. Most of all, she saw a need for us today to draw nearer to the one who gave His all in all, and that one is Jesus Christ. Because of this, she would write:"
I am Thine, O Lord, I have heard Thy voice,
And it told Thy love to me;
But I long to rise in the arms of faith,
And be closer drawn to Thee.
Consecrate me now to Thy service Lord,
by the pow'r of grace divine;
Let my soul look up with a steadfast hope,
And my will be lost in Thine.
O the pure delight of a single hour
That before Thy throne I spend,
When I kneel in prayer, and with Thee, my God,
I commune as friend with friend!
There are depths of love that I cannot know
Till I cross the narrow sea;
There are heights of joy that I may not reach
Till I rest in peace with Thee.
Draw me nearer, nearer blessed Lord,
To the cross where Thou has died;
Draw me nearer, nearer, nearer blessed Lord,
To Thy precious bleeding side.
It was this verse that Fanny had in mind when she wrote these words:
"let us draw near with a sincere heart in full assurance of faith, having our hearts sprinkled clean from an evil conscience and our bodies washed with pure water."
This is a beautiful version with piano and violin.