I know that some beers have less carbonation than others. But does anyone know of a beer that is not even meant to be carbonated?
Unblended lambics are flat, and some of the other Belgian sours can also have very low CO2, but I don't think that's what you're going for.
I'd go with the other poster's suggestion and consider cask-conditioned English ale style ales. I particularly like the lower-alcohol styles since they're much more sessionable; you don't wake up the next day feeling like something died in your mouth. Don't be put off by 2-4% ABV, you can pack a LOT of flavor into those little guys.
Also, since you're homebrewing, there's nothing that says that you have to carbonate your beer if you don't want to. It will mess up your perception of aroma and might make the beer seem sweeter or more watery, but presumably you can compensate.
I'm skeptical that CO2 has any effect on hangovers. CO2 mostly comes out of solution (hence beer belches) and I don't think it's absorbed through the lining of your stomach or esophagus. By the time the beer gets to your small intestine, I'd think that the CO2 is mostly gone. The big culprits in hangovers are ethanol, as well as the higher alcohols, ketones, esters and other minor flavor/aroma compounds. Those, plus dehydration as your body flushes these chemicals and their metabolites out of your system.
I think that there was some other factor in your uncarbonated beer that made it less potent hangover fuel: lower attenuation = lower ABV, lower fermentation temperature = fewer fusel alcohols & esters, different yeast strain = smoother fermentation profile. You'll need to look carefully at your "yeast management" and fermentation techniques to figure out exactly what's going on.