Metal bands that don't scream...hmm...most of the stuff in Opeth's songs is screaming, but they break into a lot of musical singing pieces during most of all their songs. They even have an all acoustic album that is pretty kickass, but it's completely different from all of their other albums. Nightwish is a pretty good band, but you really can't go wrong with Lacuna Coil or A Perfect Circle.
Screaming is overrated. At least, it's poorly done too often nowadays to be taken seriously. OP is better off looking for older material. Bands like Old Man's Child, Moonspell, Dream Evil, Opeth, and every Tristania album apart from Ashes would work. Let me sift through my list and find bands without screaming that haven't been mentioned yet...
Helloween
Hypocrisy
Katatonia
Lacrimosa?
Lame Immortelle?
Moonlight
Passenger
Sentenced..?
Sirena (MAYBE. MAYBE.)
The Gathering
The Machine in the Garden
Tiamat
XIII Stoleti
Errrr.. OK at this point I'm trying to figure out what's Gothic Rock, what's Metal, what's Hard Rock, what German Rock, what bands no longer exist, and what's Russian Techno disquised as any of the aforementioned genres so I'll stop.
Check out Pig Destroyer anyway. //content.invisioncic.com/y282845/emoticons/satan.gif.9c6a335ed7aeeed3ee273e573f1fcaac.gif
ACDC, Metallica, Black Sabbath/Ozzy, Disturbed, Rob Zombie, Godsmack, System of a Down (this one maybe not so much but I still like them even though I hate Death metal), Velvet Revolver, Breaking Benjamin, Motorhead, Alice in Chains...theres more, I'll add as i remember them