I have the same problem

Check the idle position sensor.

sounds like a bad throttle position sensor or a fuel pressure regulator. also check the idle air control vavle

Try this. When you first try to turn your engine on, keep your foot on the gas pedal. The car should turn over but not start. (don't worry, its not supposed to). Take your foot off the gas and turn the ignition on again, but Don't touch the throttle this time. That should set your choke correctly. To summarise: Turn key with foot on Gas. Take foot off Gas, and turn key again.

Some models have to be retaught how to idle once the battery dies, so you need to start the truck and feather the gas peddle to keep it running and do this for a few minutes, the computer has to relearn how to idle the engine.

While the engine is cold, try starting it and giving a little gas while you start it. If it starts, release your foot and see if it keeps running. If it stalls out, start it again (by giving a little gas again) and then keep your foot on the gas slightly to keep it from going to it's normal idle. Once the engine is warmed up, let off the gas and see if it stays running again. If it does then you need to clean/reaplace the IAC valve that is on the throttle body.

If you can, remove it and clean it out with carb cleaner (like B-12) and an old toothbrush. It could save you some money from having to replace it.

ok i have a 1998 durango with the 5.2L 318 engine. had trouble with a bad cell in my battery. I went out bought an optima red top battery for it 800cca. put it in and had same trouble. No start without putting foot on gas. wouldn't stay running without holding foot on gas. CPU has to relearn it's programming once you unplug it for long periods of time. mine was without power for about 2-3 weeks till i got new battery. Start with this.. first check under air filter make sure a squirrel hasn't made a mess under your air filter and it's clogging the air intake. Then pull off the lid to your injector cover ( the screw down lid coming from the air box ( with air filter in it) to "carb") make sure all your cables move freely. put that all back together and close hood. Start truck with foot slightly on gas pedal to get started. keep it at about 1k to 1500 rpm for about 1-2 minutes. Rev motor to about 2500rpm and let it die out if it will. Get out lock doors with key fob if you have it. if not make sure to lock doors. ( trust me it worked for me) let sit for 5 minutes. come back out and unlock door and relock door 3 times with fob ( if you have it ) or key if you don't have a fob. Then unlock it with the key and enter vehicle. Start truck as before and keep it around 1k to 1500 rpm again for about 30 seconds rev it up and should keep running. Mine did. I think the iac was stuck from sitting and has now since loosened up.
Good luck hope ti works for you.

The idle relearn needs reset. Keep your foot on the gas at about 900 rpm. Make it idle for about 5-6 minutes. Slowly ease up on the gas until it will idle on it's own, it might take a while. After you get it to idle, it might be idling real low, but on its own, Let it sit and idle until for 20 minutes. After that it should be back to normal. You will have to do this every time you have the battery unhooked for a while.

Have you pulled the code with a OBD II reader ? Please let me know which codes you find.

Will it stay running if you keep your foot on the gas? Or is the compressor locked up not allowing the engine to turn?

TRY TURNING UP YOUR IDLE ALSO have it tuned up not enough spark will cause ths prob.