Please tell us more about your engine so we know what we're dealing with - carbed? EFI? In any event, the catalytic converters won't solve your mixture problem - if I were you I'd work on that separately and try to get the engine working with reasonable a/f ratios before you put your cats on.
The emissions systems on cars are designed to work in concert -- all the pieces work together. Generally speaking, the EFI uses feedback from O2 sensors to keep the air fuel ratio in a pretty small range - so that the catalytic converter has the capacity to catalyze further oxidation in the exhaust stream AFTER the products of combustion have left the combustion chamber. By carefully controlling the mixture with efi and feedback from the O2's, the amount of harmful emissions the cats have to clean up is limited. Specifically, there are oxides of nitrogen (NOx), unburned hydrocarbons (HC - if you're rich, you've got lots of these), and carbon monoxide (CO) - these are the 'harmful' controlled emission products. The emissions systems are designed to reduce these. The cats are constructed in a way that they speed up additional chemical reactions when the harmful emissions (above) pass over the ceramic matrix inside the cats - turning most of the harmful HC, CO and NOx into harmless CO2 and H2O. In order for those chemical reactions to occur, excess oxygen needs to be present in the exhaust stream - and that's what the air pump does - it provides the excess oxygen necessary for the cats to do their job by pumping air into the exhaust system. Matt90 - the airpump is DEFINITELY there for the cats - both components work together to reduce emissions. But here's the thing - the cats don't work so well when the mixture gets a long way off from what they were designed for, and they also don't work well when there's no excess oxygen for the chemical reactions the cats support. So, cats without excess oxygen (air pump) and mixture control may not work like they're supposed to. EDIT - as cars have gotten newer, and fuel injection systems and combustion chamber design has gotten even better at controlling the mixture and resulting products of combustion, many newer cars have been able to do away with the need for the air pump. But that doesn't mean it's not required to optimize an older system.
So, just throwing a set of cats on thinking it's gonna help "burning lungs and shortness of breath" probably ain't gonna work - it's likely only to result in burning lungs, shortness of breath, and a shortness of cash in your wallet.