Toyota exhaust problem HELP!

My 88 Corrolla All Trac is having some kind of exhaust leak problem. There are no visual signs but here are the symptoms:
1. I am getting headaches and eye strain when I drive this car. Not when I drive others. I am dizzy for sometime afterward.
2.The car "sputters" when down shifting or accelerating, BUT NOT from the engine. It literally sounds like the car is farting through the exhaust.
3. The engine runs great! I am getting approx.23 mpg combined and it starts right up. Very smooth. I have had the manifold gasket replaced, plugs, distributor, wires and rotor. There really seems to be nothing wrong with the ignition and I don't smell and gasoline from the injectors. They seem very clean.
So...
Why is it "sputtering" or backfiring in a continuous sound,,
And,
What about what I am assuming are fumes. I am about to get the catalytic converter replaced out of frustration but uncertain this resolve the issue. Only one mechanic has said he doesn't believe it is an engine problem and said that after these cars have run rough (two of the plugs had pracitcally fused and the rotor was all black from carbon before I replaced them) for a while the exhaust systems could get messed up. Any help would be extremely appreciated.

Welcome to the AF!

Sounds like your muffler or cat converter may be clogged up and exhaust backing up and leaking - causing your headaches etc. If the muffler looks in good shape, then it's probably your cat converter.

Welcome to the AF!

Sounds like your muffler or cat converter may be clogged up and exhaust backing up and leaking - causing your headaches etc. If the muffler looks in good shape, then it's probably your cat converter.

Thanks Brian,
But wouldn't the car run sluggish? It runs like a top, but it is running hot, and I am getting that smell. I have suspected the cat for sometime as the car came with lots of records, including a muffler change, but the cat might just be original. The Meinike guy who did the free inspection did not have his thermometor to test it, but said that if the cat was bad the car would be running sluggish. Do you think the car could still run well but the cat could be trouble.
Thanks again for the welcome and the reply!

I guess it depends on how plugged it is. If your car can drive 90mph with gas to spare, then no way anything is plugged, but if it's only partially plugged, then maybe it would run fine up to a certain point, then not run as well above that.

You say it's running hot. That would imply cooling system or head gasket. Is the temp gauge above normal or are you going by some other factor?

Temp gauge is in the middle. But the engine seems to be very hot when I shut down. Rad was replaced a year ago, so maybe it normally runs this way. Do these things have heater cores like the old cars. I still think your point about the cat maybe close. When I shut down, I hear a "pop" from the exhaust, like a piece of metal that was too hot expanding, than contracting.

unbolt your cat and check the top of it. i had similar problems with the sound, and all the backfires ended up putting holes on my cat. hopes this helps

u'd notice if the cat wos blocked, the back pressure would b hold the engine back a fair bit with under hard throttle, throttle response would b considerably affected. have u checkd if the gaskets on the exhaust flanges? a quick ez way 2 diagnose a exhaust leak, cover the tail pipe with ur hand, hold it, if engine slows but doesnt stall or splutter, then most likely theres a leak, next is just move ur hand across the exhaust system n feel for the leak. however, some systems have 2 layers of piping, if the inner layer is corroded n leaking, its hard 2 locate the location.

Temp gauge is in the middle. But the engine seems to be very hot when I shut down. Rad was replaced a year ago, so maybe it normally runs this way. Do these things have heater cores like the old cars. I still think your point about the cat maybe close. When I shut down, I hear a "pop" from the exhaust, like a piece of metal that was too hot expanding, than contracting.

If your temp gauge is in the middle, you're good.

As was posted above, try and find the leak and/or the plug. If you can't, I don't think you have any choice but to bring it to a muffler shop and ask them to find it and repair it. Get more than one opinion. Make sure they aren't just trying to sell you a whole exhaust system when all you need is a muffler or cat converter. Have them show you the leak - keeping an eye out for the possible screwdriver-through-the-muffler technique of finding a leak. Watch them like a hawk.