...selling it or parting it out? Blue book value is $4400. Body damage is crashed front end and crashed back end (headlights and taillights are housings are broken but all lights work). Also has oil & radiator leak, needs new ignition switch, needs brakes, broken windshield, 3 bald tiresand one flat tire, no spare, no cigarette lighter, no stereo, no turn signals, broken antenna, missing weatherstripping on roof.

Parting it out will get you LOTS more money :)

Sure, If you're willing to wait a lot longer for your money.

Also, having the car sitting there rotting will not be good for any remaining good parts. I'd remove any nice stuff (leather seats, engine, good glass) and store them properly, then sell the rest to a crusher.

Even then, you might still be better off selling it to a junkyard. At least you'll get all the money at once.