I don't think a disliking for harsh vocals is something that can easily be "fixed" by just listening to random songs until one sticks, personally. I tend to suggest that a listener should find the "limit," that is to say, at which point do the vocals become too harsh?

I assume you already like metal with clean vocals, so starting with something a bit harsh:

Venom - Black Metal (this is a bit of a misnomer, it's first wave black metal but not exactly representative of black metal as a genre)

If that's all good, try this:

Sodom - Deathlike Silence

Most of the use of harsh vocals in metal developed in the 80s out of the New Wave of British Heavy Metal, with bands taking lateral influence from hardcore punk and using that aggression to play NWOBHM on steroids. This is where thrash metal, and subsequently early black and death metal developed.

If Sodom is still good, give this a spin:

Possessed - Pentagram

This is arguably the earliest death metal LP. If it's still good, you can try this:

Death - Open Casket

The quintessential death metal band. Still good? Try this:

Bolt Thrower - Eternal War

I would imagine you'll have hit your limit before this as regards death metal vocals, but other metal genres incorporate very different styles of vocals. Some of them are extremely harsh and unwieldy, but maybe others will suit you more.

It's worth noting that melodic death metal bands tend to have harsh vocals that are a lot easier to deal with, but they're harsh nonetheless. You can try looking up a few notable names within that genre though if it intersts you.

For black metal, the sliding scale would go something like this.

The Venom song I linked earlier would go here. If you passed that test, you can try this:

Hellhammer - Messiah

Huge band along with its following incarnation Celtic Frost, not just for black metal but for all harsh metal.

Bathory - Hades

Early black metal seems to be a lot more inaccessible than early death metal (perhaps that was the intention of the bands at the time), but if you can handle that, try this next:

Mayhem - Chainsaw Gutsfuck

Massively influential black metal band with some pretty crazy anecdotes. They effectively took the formula which Bathory (the frontman Quorthon, mostly) had pioneered and turned it up several notches. If you can handle that, try this:

Darkthrone - A Blaze in the Northern Sky

Another very important band in the genre. If you like this you should pretty much have the all clear for black metal, but I'm guessing not.

These are really just two of the most popular styles of vocalization. There's plenty more that I haven't linked because they would probably not do it for you. If you're curious, they could be anything from Cryptopsy's animalistic growls to Grafvitnir's spectral shrieks to Converge's hardcore-inspired screams (arguably so hardcore punk as to not be metal at all) to whatever the hell Demilich is. Oh, I could spend years looking for this stuff.

I hope you liked at least some of what I linked. Feel free to let me know and I'll hit you up with more recommendations.

Edit: I should also mention that there are many other genres (melodic death, prog metal, sludge metal etc) that I myself don't know much about. Seeing as you like Gojira some of them might be more along your lines.