In my experience, gels work best - they are slippery and don't dry out as easily. I've tried creams and soaps and they just aren't as good. Soaps are awful. I think classic shaving folk use them to make themselves feel classy, not because they are actually better. As you've found out, fragrance and dye can cause irritation, avoid them.
Aveeno has a shave gel which I've found very effective. It's a little more expensive but some places sell their own knockoffs. Rite-Aid has a very, very convincing imitation. Today I bought a $2 can of Noxzema shaving cream that has dimethicone in it, the skin protectant found in products like Blistex. It does have some fragrance, but it might not matter. I haven't tried it yet.
You might also try shaving oils. I tried them once and found them to be okay, if a little messy. The bottle says 1-2 drops for your whole face, but I needed more like 5 or 6. I've read about people using a little bit of oil as a sort of undercoat, using a cream or gel on top of it. That might work well.
It also helps to shave right you shower, as the water softens the hair and skin.
Oh, here's something interesting. I tried one of those depilatory creams a while back. These creams contain an enzyme that breaks down keratin, which hair is made out of. It irritated my face a little and didn't do a very good job removing all the hair. Going against the advice of the label on the tube, I used a razor and shaved after using the cream in order to remove the excess hair. I shaved against the grain and had NO irritation or redness. My hypothesis is that the cream also softened the skin, because skin contains keratin as well (which is why you can't leave it on your skin for too long). It was as if the top few layers of skin cells just slid right off with the razor, leaving no irritation. I think I might try mixing some of this cream with a shave cream and letting it sit for a few minutes before each shave. Should soften the hair and skin quite a bit.
Bassoon, n. A brazen instrument into which a fool blows out his brains.