2003 Dodge Durango (5.9L V8) won't stay running unless you keep your foot on the gas.

SOURCE: Hello , I have a

P0305 code is a cylinder #5 missfire. The codes last digit is the effected cylinder, i.e. P0305 is cylinder #5 and a P0302 is cylinder #2 and so on. A P0300 is a multi-cylinder missfire.
Thoery 1: Ignition system! If the cylinder is being injected with fuel and not getting spark, the un-burned fuel would simply be pumped out the exhaust system and can be the source of the fuel smell. Check the ignition system and the firing order. I know you said you already did this, but I have to include this for other people in the future who might be using this to help them with a simillar problem.
Theory 2: Bad injector! If #5 cylinder injector is stuck at full fuel (leaking badly due to an internal mechanical failure) it will be pumping way too much fuel into the cylinder for the cylinder to burn. The outcome is the same as before, it just pumps the raw fuel out the exhaust system. Just as before, compare 2 cylinders by inspecting the spark plugs of #3 and #5. If an injector is stuck, that cylinder spark plug will be soaked with fuel. Injectors cannot be repaired, just replaced.
Theory 3: Low compression! If #5 cylinder is low on compression it will not fire the fuel as it is injected. A simple compression test is needed to tell if there is a mechanical failure internal to the engine causing low compression.
Theory 4: Bad intake plenum gasket! The intake manifold has a steel plate bolted onto the bottom of it with a gasket called a plenum. This plate is inside the lifter valley. If this gasket fails it will cause a vaccume leak into the intake manifold that will create a lean mixture condition for the cylinder that is nearest the failed spot. This lean mixture does not light or burn effectivly and can cause an unburned mixture to enter exhaust manifold causing the same sympoms as above.
Obviously this problem needs further diagnostics to pin-point the source of the failure. These are simply starting points and there can be many other things that can cause this type of problem, but these are the most likely. Hope this can be of some help, have more questions? Just ask!

SOURCE: Why won't my truck stay running when I'm not

Try cleaning your throttle body. Over time on high mileage vehicles, the throttle body butterfly valve will gunk up with sludge, and dirt. This will prevent air from entering the intake manifold while the engine is idling, causing low idle/stalling. A can of throttle body/carb cleaner will do the job. First, remove the intake tube to get to the butterfly valve. Spray a small amount of throttle body cleaner directly into the intake, and wipe any excess fluid with a cloth. Also, inspect your air filter. I hope this helps
Matt