The Sacred Harp is full of these types of songs & we sing them even when there is no funeral because they are beautiful. It does contain some that were really 19th century standards.

Isaac Watt's Why do we mourn departing friends to the tune China, by Timothy Swan is found in many 19th century American hymnals. I've actually found 2 clear parodies in the college songbook Carmina Collegensia (1868)

Even more common as words, but apparently many different tunes are these words by Alexander Pope

THE DYING CHRISTIAN TO HIS SOUL
   
VITAL spark of heav'nly flame!   
Quit, O quit this mortal frame:   
Trembling, hoping, ling'ring, flying,   
O the pain, the bliss of dying!   
Cease, fond Nature, cease thy strife,         
And let me languish into life.   

Hark! they whisper; angels say,   
Sister Spirit, come away!   
What is this absorbs me quite?   
Steals my senses, shuts my sight,   
Drowns my spirits, draws my breath?   
Tell me, my soul, can this be death?   

The world recedes; it disappears!   
Heav'n opens on my eyes! my ears   
With sounds seraphic ring!   
Lend, lend your wings! I mount! I fly!   
O Grave! where is thy victory?   
O Death! where is thy sting?   

I've found many copies of the words in online hymnbooks. References to compositions setting these words included Edward Harwood (late 18th or early 19th cent. English); Havergal Brian (UK 1924); and Claremont by Merrill and Temple (US 1799)