pdq, I can't see enough similarity with the Foster song.
There are a couple of songs about "Gentle Annie," but they are about her death, which rules them out.

A song with the same idea is "Meet Me Maggie," by Will S. Hays, but it is not the right song. I'll quote it anyhow.

MEET ME MAGGIE
Will S. Hays, 1874

Chorus.
Meet me, Maggie, gentle Maggie,
I'll be waiting dear, for you,
Oh, my heart is lonely pining
For your smile so warm and true.

1
Meet me, Maggie, in the springtime,
When the flowers bloom again,
When each songster loves to warble
Some melodious, happy strain,
Meet me when the twilight shadows
Gather round the brow of day,
When the smiling face of Nature
Looks so blooming and so gay.
2
Meet me Maggie, in the meadow,
And I'll tell you where I've been,-
Where in Erin I have wandered,
All the sights that I have seen.
First I'll tell you how I love you,-
How I missed your sunny smile;
Then I'll kiss you and tell you
I was true, dear, all the while.
3
Meet me, Maggie, darling Maggie,
How I wish the day had flown!
When the little birds are sleeping,
Come to welcome me, my own.
Nevermore shall we be parted,
For my heart shall never rove,-
Like a weary bird it wanders
Back to you, my only love.

1874, published by J. L. Peters, NY, sheet music in guitar arrangement. Lester S. Levy Collection, http://levysheetmusic.mse.jhu.edu/levy-cgi/display.cgi?id=133.123.000;pages=3;range=0-2