I forgot where I read/heard this, but it was something along the lines of using a loaded sling as a club if you are closer to the target. Is this historically realistic? Or was it just an idea that someone had?

i agree with Kuggur slingdog, i think just a stick you find on the ground or a large rock could be used more effectively as a melee weapon than a sling. hence i would recommend a retreat before facing any kind of melee infantry. also the slingers were pretty light infantry, so it should be pretty easy to escape armored soldiers.

A slinger flailing his sling around wouldnt have a chance against a swordsmen etc.

well , if you get yourself forced into a corner i supose you could sling untill the last minute, then use it once like a club. but after that you had better have a knife or what ever handy, cause then  your hip deep in poo.

Maybe you could grab both ends of the sling, if it's sturdy enough, and use it two-fisted like one fights with a belt (use it to entangle, parry and deflect, aim for a snap under the chin, hope for a strangle)-- but against an armoured,  heavily armed fighter with edged weapon, blunt impact weapon, or polearm ? Hmm.

It wasn't not just because of folklore that David is described as carrying a staff in addition to his sling. As a shepherd, he knew that the sling would be rather useless when he failed to hit a predator at distance and it got within claw or paw reach. It was then that he probably would have used the staff to try to keep at least a minimum distance between himself and the animal (or, maybe, human thief). The staff may have even been pointed at one end. Of course, a lance or spear or sword would have been better, but these were comparatively expensive. I'm sure he also carried a knife as a last option, as every man or boy in that time and others. Why would he have taken his staff into the duel with Goliath if not as a possible second weapon? For a quick getaway after missing Goliath's vulnerable spot, it would have only been an obstacle.

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Actually it would make a very good weapon against a well armed foot soldier.  
It will strike over and behind a shield, can be used to entangle a sword or sword arm and strike a hard blow against the arm at the same time.  
 
It might not have been the close quarters weapon of choice, but  a flexible club can be a very effective weapon.  

In some cases it would be good to use it as a club if somone was behind a sheild. But since your close enough to the enemy, you might as well shoot him with it because its usually really easy to sling a rock at somone at point-blank range.