Ballads, Lyrics and Hymns can be found at Ballads, Lyrics and Hymns - Making of America Books, as also is Early and Late Poems of Alice and Phoebe Cary and The Poetical Works of Alice and Phoebe Cary. (On a quick scan of the contents I didn't see The Dying Mother in any of them, but a more detailed scan of the books might be more revealing).

At least 2 of her poems were set by Joseph Philbrick Webster, but this one is not listed there.

Mick