Weird request because there's no real solid definition of what defines "alternative," but I generally think of the post-grunge styles that became big. Or you can find heavy alternative bands that aren't doom but that have slow, depressive songs.

Something like this

http://www.youtube.com/watch?v=NLQq8JJXmuQ
Fear Factory— Descent

http://www.youtube.com/watch?v=EU4L6THYAbM
Soundgarden— 4th of July This would have to be one of the best doom metal songs ever

http://www.youtube.com/watch?v=vc4off_b4Qs
Deftones— You've Seen The Butcher

http://www.youtube.com/watch?v=QAZcFVuSo0M
Alice in Chains— Dirt

http://www.youtube.com/watch?v=C_HAgFs6I4o
Lacuna Coil— Losing My Religion(REM Cover) This band's already alt-gothic metal so this may be cheating

http://www.youtube.com/watch?v=6pyUO2m17gU
The Melvins— Heaviness of the Load

http://www.youtube.com/watch?v=trinU3VD1Zo
Chevelle— The Red

Some of Ministry's "Filth Pig" album should fit the bill:
http://www.youtube.com/watch?v=QloZREyA8Xk

And this is a bit of a wild card, but check out the song "P.C. Keen" by folk/punk band The Levellers; totally unlike anything else they ever did:
http://www.youtube.com/watch?v=VT6-B_7WY6I

This would be a good addition as well.

https://www.youtube.com/watch?v=cEDSDZ6mJcg
Alice in Chains— Rain When I Die

https://www.youtube.com/watch?v=xuGwk0VDYYQ&feature=kp
Rage Against the Machine— Settle For Nothing

I really think it's quite a weird request. I kind of feel like understanding what you're searching for. But since doom itself has already such big room for different styles and alternative rock/metal is often influenced by pretty much everything, it's hard to tell where the alleged doom metal-influence starts and when it's just a heavy ballad.

A Perfect Circle definitely have some songs that may suit what you're looking for (Sleeping Beauty, Thomas, Breña, Pet) and I'm sure I could name more if you'd narrow down the specific sound you're searching for at least a little more.

How did I forget? Even though I HATE the album, what about:

My Dying Bride - 34.788%... Complete

and maybe also

Type O Negative - Life Is Killing Me

When I think of alternative doom metal, those are the albums that come to mind.

Necro'ing the thread in case any other users have something to contribute.

I knew Ghost Brigade would fit here, mentioning R.E.M. was the last nail in the coffin. Their alternative doom sound pretty much comes from a combination of Katatonia and post-rock, with growls/screams thrown in (not sure if you mind them or not). Their new album will be released tomorrow and it's amazing! (This song is older though, still a lack of material on Youtube.)



EDIT:

Eggnog - The Sloth

Uriah Heep - Rainbow Demon

My favorite doomy Soundgarden tracks are 4th of July, Limo wreck and Slaves & bulldozers.

No "Beyond the Wheel" or "Toy Box"? 4th of July is probably the best doom-oriented Soundgarden song though. I think that's unanimous.

Life of Agony's heavy as fuck debut. A ferocious blend of NY-hardcore and doom.



A Pale Horse Named Death. Formed by Life of Agony/ex-Type O Negative drummer Sal Abruscato. Sounds halfway between Alice in Chains and Type O Negative minus the synthwork.



Seventh Void. Former Type O Negative's axeman Kenny Hickey's current band. Also features Johnny Kelly (ex Type O) on drums. Sounds more like Alice in Chains with Chris Cornell on vocals.

You haven't lived until you heard their tracks "Loud Love"
https://www.youtube.com/watch?v=j0xosEFe8Ik&app=desktop

and "Nothing To Say."
https://www.youtube.com/watch?v=Uhke_1WnX-Q&app=desktop
And speaking of grunge, I discovered Nirvana's track, Big Long Now, very sludge doom, very alternative.

https://www.youtube.com/watch?v=6PL3HVCreWM&app=desktop

Gothic riot grrl band Jack off Jill brought us "Surgery." Unexpectedly sludgy and plodding alt-doom.

https://www.youtube.com/watch?v=ORr1riA3HEE&app=desktop

Mindrot falls into this category a lot of the time , particularly their second album "Soul" :

I wouldn't categorize "Loud Love" as doom myself. It's more of a straightforward heavy metal song with some stoner rock kind of bendings here and there. "Nothing to Say" is fairly doomy I guess.

I kinda feel the band Across Tundras belong in this, you might enjoy them if you haven't given them a listen

random track:
https://www.youtube.com/watch?v=SfAsFJPLcYM

This is a good one.