Okay, zero oil pressure is not good to your engine... Well, same thing happened to me before. I have a 318(5.2L) Dodge Ram, and your R/T should be 360(5.9L), but the construction of engine is pretty much the same push-rod V8.
The cause of "sudden zero-oil-pressure" at stop or 'when I shake the engine' was because of "sludge on oil pick up screen." I believe you got the same oil pick up... and you gotta clean it up ASAP.
Apparently, Chrysler's 318/360 engines are known to create lots of sludge, so I'm pretty sure that you have sludge issue.
I've done this "cleaning" twice, and before last time I put the oil-pan back, I had missed putting silicon on gasket at some spot, and a lil dirt got into the pan, and those dirt+bit of sludge again clogged the oil-pick up screen. Make sure that your oil pan gasket is fit and siliconed enough.
Well, because you are not getting oil pressure, you may have damaged piston-rings or valve-seals, that is making your engine to burn oil.
Secondary, your transmission. If it is an Automatic, Chrysler's Automatic on Ram and Dakota are known to have "valve-body" issue, which does not keep the fluid pressure on top of transmission. You may want to check on that also. (well, I had the same problem with my Dodge Ram).
Good luck, and check up your oil sump screen ASAP. Dakato R/T is a very nice truck