Here are some selections from a list that I have, for reasons, in no particular order:

“These Foolish Things (Remind Me of You)” - Ella Fitzgerald

“All the Things You Are” - also Ella Fitzgerald

“Hopelessly Devoted To You” - Grease soundtrack

“Something” - the Beatles

“Nothing Else Matters” - Metallica

“Wishing You Were Here” - Chicago

“Heaven When We’re Home” - The Wailin’ Jennys (Maybe not exactly a love song, but oh well.)

“Iowa” - Dar Williams

And one more for lulz:

“Friendly Duet” - Flanders and Swann

~Neshomeh

But I happen to have one in my brain's music repository which may be of the type you seek.

—doctorlit, dropping a sad all over this Board

It's all because I used to be a huge hipster who only ever liked jazz. You gotta like sap if you're going to listen to that stuff.

My favourite song of all time is still Chet Baker's Time After Time, and all his versions of it are good but this--this one incredibly specific one that I can't, for the life of me, find anywhere other than Youtube (I think it was live at Boston, 1954)--is my absolute favourite: https://www.youtube.com/watch?v=00eq3Uh1hPE.
Chet Baker's entire oeuvre is ridiculously romantic--the guy's got sap coming out his pores. His versions of My Funny Valentine, My Buddy, and While My Lady Sleeps (bit neckbeardy of a title there, innit? There's a set of words utterly ruined.) are all very bloody good and very bloody sappy.

Almost all of Gershwins songs fit this template, especially when they're sung by Louis Armstrong and Ella Fitzgerald. I'm pretty fond of Our Love is Here to Stay. It's kinda cheery sounding, actually, but there's such a strange... genuineness to it? Something about it got to me. Maybe I was just in a strange mood at the time.
Louis Armstrong's Give Me A Kiss To Build A Dream On is also fantastic--much slower and more, I guess, classically sappy. There's a weird... Pixar feel to it. It feels like a song Pixar would have used in their heyday. It's my go to 'oh that's a super romantic sappy song right there' that I jump to in my head.

Madeline Peyroux's version of La Javanaise has very much the same feel. It's just... it's so romantic, dude, you don't even know. I'm, personally, a very un-romantic person, but it's the sorta song that fills you with a longing for that whole fake romanticised version of Paris where it's all gentle lights and people dancing and there's no racism or anything of that kind.

I've also got Same Drugs by Chance the Rapper. It's not actually literally about drugs that's like a metaphor. It's got Peter Pan metaphors and stuff too, it's wild. And, uh, very tender.

It's kinda weird to say considering the bloke's reputation and the fact that he's completely nuts, but pretty much any of Kanye West's songs dedicated to his family are so utterly genuine and sappy (maybe love song is a bit of a stretch, here?) that it's almost easy to forget he's, uh, completely nuts.

Violent Crimes is him sort of, reflecting on his own misogyny, and the misogyny of men in the world in general in the face of his growing daughter admitting that--while it's irrational--he doesn't want her to grow up, become a woman, because he's terrified for her.

He's also got Family Business (about his family, obviously,), Only You (about both his mum and his daughter. He sung it with Paul McCartney, lucky guy, I'm sure he'll blow up after this) and, uh, probably a couple other ones. Guy likes his daughter I guess, what can I say.
Those songs probably have swears in them by the way, if you wanted to watch out for those.

*reeeealy deep breath*

If My Heart was a House - Owl City
Call it Love - Blackmore's Night
Village Lanterne - Blackmore's Night
Just Call My Name - Blackmore's Night
I Have Been Around the World - Dar Williams
Amarantine - Enya
Solace - Enya
Fallen Embers - Enya
If I Could Be Where You Are - Enya
Coin Laundry - Lisa Mitchell
Clean White Love - Lisa Mitchell
Where You Are - Lisa Mitchell
Spiritus - Lisa Mitchell
Better Left Unsaid - Lisa Mitchell
Walk With Me - Lisa Mitchell
Diamond in the Rough - Lisa Mitchell
Know You're Somewhere - Lisa Mitchell
Pirouette - Lisa Mitchell
Valium - Lisa Mitchell
Monsoon - Rachel Rose Mitchell
We Can Only Listen - Ruth Moody
Dancing in the Dark - Ruth Moody
Nest - Ruth Moody (this is about as adorable as it gets)
One and Only - Ruth Moody
Valentine - Ruth Moody
Closer Now - Ruth Moody
Mona Louise - Wailin' Jennys
Asleep at Last - Wailin' Jennys
Firecracker - Wailin' Jennys
Mornixuur - Bel Canto
Night Sky - CHVRCHES
New Way - Zoe Gray
Party of One - Brandi Carlile
Except You - Zoe Gray
Kissing on the Boardwalk - Zoe Gray
Monody - TheFatRat

I...think that's everything?

* On A Sea Of Fleur-De-Lis (Richard Shindell)
* Forty-Five Years (Stan Rogers)
* Dimming Of The Day (Richard and Linda Thompson)
* River (Joni Mitchell)
* Eleanór na Rún (trad. Irish)
* The Water (Johnny Flynn and Laura Marling)

And now, my personal favourite love song:
* Northern Sky (Nick Drake)

I hope these help with whatever you're looking for these songs for.

*Thousand Years - Christina Perri
*Thinking Out Loud - Ed Sheeran
* Marry Me- Train
*Bubbly - Colbie Callait
*Falling For You - Colbie Callait
*The Rhythm of Love - The Plain White T’s
*1234 - Plain White T’s
*Marry Me - Train
* Rock Lobster- B-52s
* 500 Miles- The Proclaimers
* Never Gonna Give You Up - Rick Astley

...Okay, I might be exaggerating on the last three :P

I have a soft spot for sappy songs. Like, a really big one. So here are a very of my very absolute favorites:

-Walking After You, by the Foo Fighters. Also, the less gentle Up in Arms and Everlong, off the same album. Especially Everlong. Who doesn't like Everlong?

-Silly Love Songs, by Paul McCartney. Seriously, why isn't this song more popular? Always puts a smile on my face.

-Maybe I'm Amazed, also by Paul McCartney. Man, Paul is good at this.

-Tears, by Rush. Off 2112, which is an amazing album. Fun fact: this is one of the very few Rush songs with lyrics written by a member who isn't Neil Peart.

-Tender, by Blur. Also Ong Ong, sort of?

-Iris, by The Goo Goo Dolls. And I know that'd a clichéd pick but I don't care.

-Romeo and Juliet, by Dire Straits

-4th of July, Asbury Park (Sandy), by Bruce Springsteen. This song is romantic, but it also just... personally means a lot. Lot of memories.

-Just The Way You Are, by Billy Joel

-Fly Me To the Moon

-God Only Knows, by The Beach Boys

Bonus mention goes to Peter Gabriel's In Your Eyes, which can (and this is true) only be played from casette on a boombox.

Which has the bonus of being a metal ballad while still coming off as tender and caring.