Difference between Sorrowful and Elegiac

SORROWFUL vs ELEGIAC: NOUN

A pentameter, or verse consisting of two dactylic penthemims or written in elegiac meter.
Elegiac verse.
A poem composed in the couplet style of classical elegies: a line of dactylic hexameter followed by a line of dactylic pentameter

Feeling, showing, or expressing sorrow. : sad.
Causing sorrow.
Full of sorrow; exhibiting sorrow; sad; dejected; distressed.
Producing sorrow; exciting grief; mournful; lamentable; grievous.
Of a person, full of sorrow; exhibiting sorrow; sad; dejected; distressed; distraught.
Experiencing or marked by or expressing sorrow especially that associated with irreparable loss

Expressing sorrow often for something past
Resembling or characteristic of or appropriate to an elegy
Of or composed in elegiac couplets.
Belonging to elegy, or written in elegiacs; plaintive; expressing sorrow or lamentation
Used in elegies
Of, or relating to an elegy.
Expressing sorrow or mourning.
Of, relating to, or involving elegy or mourning or expressing sorrow for that which is irrecoverably past.

A succession of distichs consisting each of a dactylic hexameter and a dipenthemim; a poem or poems in such distichs: as, the Heroides and Tristia of Ovid are written in elegiacs. See I.
Expressing sorrow or lamentation: as, elegiac strains.
Belonging to an elegy, or to elegy; having to do with elegies.
Verses or poems consisting of elegiac distichs are called elegiac verses or poems (elegiacs); poetry composed in this meter, elegiac verse or poetry (the elegy); and the writers who employed this verse, especially those who employed it exclusively or by preference, are known as the elegiac poets. Elegiac verse seems to have been used primarily in threnetic pieces (poems lamenting or commemorating the dead), or to have been associated with music of a kind regarded by the Greeks as mournful. Almost from its first appearance in literature, however, it is found used for compositions of various kinds. The principal Roman elegiac poets are Catullus, Tibullus, Propertius, and Ovid. In modern German literature the elegiac meter has been frequently used, especially by Goethe and Schiller. Coleridge's translation from the latter poet may serve as an example in English.
In ancient prosody, an epithet noting a distich the first line of which is a dactylic hexameter and the second a pentameter, or verse differing from the hexameter by suppression of the arsis or metrically unaccented part of the third and the sixth foot, thus:

Mourning, Heartbroken, Weeping, Tearful, Brokenhearted, Dolorous, Grieving, Anguished, Plaintive, Sorrowing, Sad, Elegiac, Heartbreaking, Heartrending, Mournful

Lugubrious, Lyrical, Rhapsodic, Ruminative, Plaintive, Poignant, Evocative, Wistful, Plangent, Poetic, Mournful, Melancholy, Melancholic, Distichs, Sorrowful

Mourning, Heartbroken, Weeping, Tearful, Brokenhearted, Dolorous, Grieving, Anguished, Plaintive, Sorrowing, Sad, Elegiac, Heartbreaking, Heartrending, Mournful

Impressionistic, Lugubrious, Lyrical, Rhapsodic, Ruminative, Plaintive, Poignant, Evocative, Wistful, Plangent, Poetic, Mournful, Melancholy, Melancholic, Sorrowful

It was so sorrowful that my eyes teared up.
John the Divine are not so sorrowful as yesterday.
When he spoke his voice was resigned and sorrowful.
Patrons modulate dollar from sorrowful walk of American drugs.
He would always look like a sorrowful white boy.
It will free your mind of sorrowful thoughts.
To this sorrowful appeal Moses answers with encouragement.
Francis Hospital, Sisters of the Sorrowful Mother, Rev.
It is sorrowful only for those left behind.
Parting is sorrowful but to choose the other over God is more sorrowful and has eternal consequences.

The OP said they did not want to call it elegiac or elegies.
You find yourself yearning to return, the elegiac soundtrack decorating your idle thoughts.
It is an elegiac poem, since dactyl pentameter exists mostly in elegies.
Busoni: The Elegiac Lullaby in F Major for Nine Instruments, Op.
Justified is charged with a logical sense of elegiac melancholy.
His elegiac effusions are all of this class.
It is composed in the Latin elegiac couplet.
Latin elegiac couplets, usually containing a moral lesson.
Lamartine is his superior as an elegiac poet.
Street Band muscle into a dignified elegiac toast.