I wondered if that might have been what you meant about "FF3." For the record, Final Fantasy IV was released in the rest of the world as "Final Fantasy II" and FFVI came out as "Final Fantasy III." The confusion lives on! Unless you're actually from Japan, you probably have Final Fantasy VI.
Okay, on those systems, you can get all sorts of Final Fantasy games, but let's focus on the ones with digital purchases: The Wii, Wii U and 3DS... and there isn't anything on the 3DS. Except Bravely Default. Bravely Default's great, it's the best "Final Fantasy" game in years, but I'm going to stick to your criteria of name-brand, numbered Final Fantasies.
The Wii and Wii U hav Final Fantasy 1, Final Fantasy 4 (under the name "Final Fantasy 2") and Final Fantasy 6, which you have. The Wii and Wii U also have Final Fantasy 4's sequel, The After Years, in the WiiWare section. In terms of emotional stories, 4 is so-so. It has a great emotional opening half, but the latter half meanders. It's not that bad but you could do better. The After Years talks a lot about family connections, I like it for that.
FF1 doesn't have a story worth talking about.
You didn't mention your PC, but you can get the real Final Fantasy 3, Final Fantasy 7 and Final Fantasy 8 on Steam. You'll need a somewhat reliable internet connection because the PC games have online DRM. Final Fantasy 3 won't fit your definition of an emotional story, it only comes close once or twice. Gexus has already recommended 7 and 8.
There are also PC versions of Final Fantasy XI and Final Fantasy XIV, the MMOs. I think I've heard the PC is the best way to play XI, but I haven't heard any opinions on XIV. Keep in mind that they are subscription based, and I don't know anything about their stories. Anyone?
If you don't mind looking for hard-copies, it wouldn't be that hard to find copies of every Final Fantasy from 1-12 for the systems you already have. Heck, multiple versions. FF2 isn't very emotional, even if it's better than 1, but I'm of the opinion that 5 is more emotional than most people give it credit. It just doesn't take itself seriously all the time. Other people have talked about the remaining games.
Keep in mind that Final Fantasy XII is only available in hard-copy for the PS2, so if you want it, you'll have to track it down! Even a PS3 won't help you there!
But if you've got your heart set on a new system, I get that. Here's some purchasing advice for the PS3 and 4:
The PSN sells copies of FF1 and 2, and every game from 4 through 9. It also sells alternate versions of 1, 2 and 4 on the PSP.
The remake of 10 and 10-2 are on the PS3 in hard copy, and of course the 13 games - XIII, XIII-2 and Lightning Revenge Returns - were released for PS3. Unfortunately, the PS4 can't play PS3 games, so that might be a dead end. Hopefully they'll be ported, but don't get your hopes up. If all else fails, 10 is on the PS2.
A few games have sequels and prequels out there that you could buy for systems you already have, but I'm sure you're not interested in those until you've tried the originals! I've mentioned a few because they're in bundles but there's more out there if you fall in love with a particular game.
I told you I had typed a lot.