Nathan Jones - The Supremes
If a woman could die of tears
Nathan Jones, I wouldn't be here
The key that you're holding won't fit my door
And there's no room in my heart for you no more
'Cause, winter's passed, spring, and fall
You never wrote me, you never called
Nathan Jones, you've been gone too long

Hit The Road Jack - Ray Charles
Hit the road Jack
Don't you come back no more, no more

What Kind Of Fool Do You Think I Am - The Tams / Bill Deal and the Rhondels
I was lost in your love and I got burned
You hurt me one time, I finally learned
So now that I know what your lovin can do
Who do you think you're talking to?

What kind of fool do you think I am?
I ain't falling for you all lover again

Run To Him - Bobby Vee
If somebody else can make you
Happier than I can make you
Run to him
My tears will dry

Arthur Alexander's and later The Beatles' "Anna" could fit the category perhaps.

That was definitely a funny episode

Tell It To The Rain - The 4 Seasons
You're crying now just like I used to cry before (rain, rain)
Turn off the tears, girl, I don't care any more
You can save all your lies, 'cause I've heard them all before
Tell it to the rain
And the stars that shine above
That it's me you're thinking of
And I'm your love
Tell it to the rain

I Will Survive - Gloria Gaynor
Go on now, go, walk out the door
Just turn around now, 'cause you're not welcome anymore
Weren't you the one who tried to hurt me with goodbye?
Did you think I'd crumble? Did you think I'd lay down and die?
Oh no, not I, I will survive
Oh, as long as I know how to love, I know I'll stay alive
I've got all my life to live, and I've got all my love to give
And I'll survive, I will survive, hey hey

Sorry, I don’t see that any of these singers had moved on.  If they’re still singin’ about ‘em they ain’t over ’em yet.  This being an oldies board, I’m sure most are over ‘em by now.
 
Adding a few more:
 
I’m Movin’ On - Hank Snow
That big eight wheeler rollin' down the track
Means your true lovin' daddy ain't comin' back
'Cause I'm movin' on
I'll soon be gone
 
At least he’s honest, he ain’t \over her yet.
 
 
Is Anybody Goin’ to San Antone - Charley Pride
Sleepin' under a table in a roadside park
A man could wake up dead
But it sure seems warmer than it did
Sleepin' in our king-sized bed
 
He’s moved out but hasn’t moved on.
 
 
Bumble Bee - The Searchers
I'm gonna have to put you down
Youve been treatin me like a clown
You know youve hurt me once before
You'll never hurt me anymore
 
Shoo-wee, you hurt me like a bee
A bumble bee, an evil bumble bee
 
 
Now baby there's no need of cryin
I'm sick and tired of all your lyin
You know youve hurt my heart again
I'm sorry baby it's the end
 
BABY?  He wasn’t over her either.

Lovers Who Wander - Dion and the Belmonts

She took my love, said she's not coming back
So my world and my sky was all turning black
Now I see the light, I'm wise to her
Cause I found that place for lovers who wander

A walkin around thinkin about her
Never thought I could live without her
She broke my heart, I really showed it
Look at me now, you'd never know it

Now my story can be told with a smiling face
I'm the luckiest guy in the human race
My dreams have gone for her
Cause I found that place for lovers who wander

Does Laugh Laugh by the Beau Brummels count?

I hate to say it, but I told you so
Don't mind my preaching to you
I said, "Don't trust him," baby, now you know
You don't learn everything there is to know in school

[Verse 2]
Wouldn't believe me when I gave advice
I said that he was a tease
If you want help, you better ask me nice
So be sincere, convince me with a "pretty please"

[Chorus]
Laugh, laugh, I thought I'd die
It seemed so funny to me
Laugh, laugh, you met a guy
Who taught you how it feels to be

[Post-Chorus]
Lonely, oh so lonely