Exhaust sputters engine light flashes car shakes

check the plugs again and see if they are oily again, if they are NOT oily check the ignition coil if it is giving current to all four plugs, if not ,change it.it might be shorting out when it is hot.........If the plugs are oily again, and assuming the valve cover has been changed,sorry to say you have an internal eng.piston ring problem. more tests must be done to pin point the problem.

sounds like you got some water in your gas ,, or your mass air flow sensor is shorting out , or dirty

Is your check engine light on? If it is yo can have it check to dee what code is stored. This should help lead you to the problem.

You have a check engine light so you know what to do now

Why would you use a car with those issues & not have a shop
at least get a start on some possible solutions

Stop using the car

You have serious cylinder misfires,so severe
you maybe ruining the exhaust converter,thus the smell

I won't write a book on all the causes or fixes,have a
repair shop diagnose

use a couple of quality injector cleaner in the tank
run fault codes
remove and clean injectors
check fuel pressure

it is not likely to be a fuel issue ... more likely it is your spark plugs or spark plug wiring. If one of your plugs is not firing right, then that piston will not function normally and cause a violent shake in the engine. This will be more noticeable when accelerating as more fuel is pumped into the piston and is not being burned off properly. This could cause the strong smell of exhaust, as the flow of the exhaust would not be normal.
It is also possible that your timing chain has skipped a tooth, causing a misfire in your engine -- this could also cause the exhaust flow to be abnormal, giving a strong smell of exhaust when the issue occurs.
Another possibility is that you have a bad fuel injector on one of your pistons ... but this is more likely to make your engine sluggish than cause it to shake violently -- usually a bad injector does not shoot enough gas into the piston, rather than pushing too much.
There is also the possibility that you have an issue with your exhaust manifold (though this is less likely and would probably make your vehicle very loud) -- if there is a hole in the exhaust that would cause the strong smell ... and the improper back pressure could cause your engine to work badly. Once again, this is unlikely to cause a violent shake in your engine ... but it is possible, so I mentioned it.
I would first check the spark plugs, then their wiring as these are the cheapest and easiest repairs that could resolve the issue. If neither of these is the problem, check to see if your car is reporting a problem code (under the passenger side of the dash there would be a blinking light -- count the number of blinks and use that number to reference an error code). If your engine is reporting an error code, that should help you diagnose the issue. If the plugs are not the problem, and the engine is not reporting an error code, I recommend taking the vehicle in to a mechanic. A professional with the vehicle in front of them will be able to provide you with a better diagnosis.

sounds like coil pack.

Checking this information car start but engine sputters while starting.

Hi:
Not sure of all what is wrong, but one thing I can suggest is to have your computer codes downloaded to see what it is showing as a fault code. Only a few codes can cause the check engine light to flash, and the most common is a cylinder misfire. Your car most likely has an ignition module or coil pack failing, and that problem is causing most of your symptoms. You can usually get your local autoparts store, such as Autozone, to scan your car for free, and tell you what is tripping the check engine light, and then you can go from there!