There are two copies at The Library of Congress American Memory Collection, with different composers. I will leave it to someone more musically literate than I to determine which tune is closer to the one commonly sung today.

They have different third verses, and some other differences in phrasing.

The 1877 version says "Words by G. P. Cook. Music by Ralph Roland. Sung with great Success by Fred. Walz of Sweatham & Frazer's Minstrels."

Note that in both copies, there is no "my" in the title or refrain.

ARE YOU TIRED OF ME, DARLING?

1. Are you tired of me, darling?
Do you wish the words unsaid,
Words that made me yours forever,
On the day that we were wed?
Tell me! could you live life over,
Would you have it otherwise?
Are you tired of me, darling?
Answer only with your eyes.

CHORUS: Tell me! could you live life over,
Would you have it otherwise?
Say! are you tired of me, darling?
Answer only with your eyes.

2. Do you miss the blooms departed
On the cheek you thought so fair?
Do you think I've grown cold-hearted
'Neath my weight of woman's care?
Life is not made up of sunshine.
Clouds obscure the bluest skies.
Are you tired of me, darling?
Answer only with your eyes.

3. Do you ever rue the springtime
When we first each other met,
When you spoke with warm affection
Words my heart can ne'er forget?
I have lived and loved, my darling.
You have been my only prize.
Are you tired of me, darling?
Answer only with your eyes.

The 1878 version says "Words by P. Gilsey Cook. Music by Fred Kenyon Jones." Lyrics are identical to those above with these exceptions:

Verse 1, Line 4: Ere my girlish beauty fled?
Lines 7-8 of each verse: Are you tired, tired of me, darling?
Answer only, only with your eyes.

CHORUS: Are you tired of me, darling?
Answer only with your eyes.
Are you tired, tired of me, darling?
Answer only, only with your eyes.

3. When life's weary task is ended,
And I leave you here alone,
Will you miss me from the hearthstone?
Will you grieve when I am gone?
Have my heart's best love been wasted?
Naught on earth but thee I prize.
Are you tired [etc.]