The Heart of the Matter by Don Henley, although that one might be about right after the relationship ends.
Also on the “Bat Out of Hell” album: For Crying Out Loud
And also from the pen of Jim Steinman: Total Eclipse of the Heart
I’m not sure if these count as “almost over” or more “should be over.”
Ex Factor by Lauryn Hill
No matter how I think we grow
You always seem to let me know
It ain’t workin’
It ain’t workin’
And when I try to walk away
You’d hurt yourself to make me stay
This is crazy
This is crazy
Break Your Heart by Barenaked Ladies
Both are always good for a little crying at the end of a relationship.
“Last Song for Shelby Jean” by David Bromberg:
I’ve got a terrible thought in my mind
I think we’ve said all we’ve got to say
And that awful hour you’ve been searching for
Well, you finally found it yesterday
Well, I think you know that it’s time to go
Not tomorrow, but today
We’ve tried in the past to make the thing last
But it only hurts longer that way
Boy, the SDMB always amazes me.
When I saw the OP, I thought of 3 songs: “You’ve Changed”, as covered by Billie Holiday;
“The Last Goodbye”, by Jeff Buckley; “Total Eclipse of the Heart”.
Why? - Annie Lennox (YouTube clip)
“If You’re Gone,” by Matchbox 20 is the first one I though of…
Annie Lennox - No More I Love Yous 
Everclear - Nehalem
You know how the bad words get around
Big noise in a hard small town
Tell me if you want me to go
Just tell me if you want me
They say you’re leaving Nehalem
They say you’re leaving me
Short of something tragic and/or violent happening I think this is one of the worst possible ways for a person to find out that they’re breaking up…
Take It on the Run, REO Speedwagon
Lyin’ Eyes, The Eagles
Don’t Talk to Strangers, Rick Springfield
Actually, I think I’ve slipped over the line into cheating songs. If those count, we’re going to have a lot more answers…
Already Gone by Blue Rodeo.
*I wish that we could find / The words to look the other way
Be careful of the things we do / And watch the things we say
Maybe then we’d get somewhere / We’ve never been before
But what’s the use in fighting / When you know you’ve lost the war
*
Strange Way - Firefall
Suspicious Minds - Elvis Presley
Smoke From a Distant Fire - Sanford & Townsend
Better Be Home Soon - Crowded House
Sail On - The Commodores
Quittin’ Time, sung by Mary Chapin-Carpenter.
Beautiful song…
Nothingman - Pearl Jam
Crosby Stills and Nash’s Suite Judy Blue Eyes
Leonard Cohen’s Hey, That’s No Way To Say Goodbye
Tom Paxton’s The Last Thing On My Mind
“End of the Road” as covered by Me First and the Gimme Gimmes.
I mixed up two songs in my head here. Never Can Say Goodbye is Gloria Gaynor, Isaac Hayes, Michael Jackson and of course, the guy who wrote it, Clifton Davis (and maybe other people).
In my head, however, I was thinking of Gladys Knight and the Pips’ Neither One of Us.
Both songs fit, I think.
The Thrill Is Gone B.B. King
also, Send In The Clowns written by Stephen Sondheim and performed to perfection by Frank Sinatra