Football shaped or similar,  does it shoot like a football or does it spin horizontally in the direction of the shot.  Or does it tumble?

I see

true. try embedding a string in one end as you cast it. it will make it stableize itself so it will fly point first.

If i recall correctly, slinging with a wide grip (the retention cord is a few fingers away from the release cord), causes the sling to roll the ammo out. As the stone naturally stabilizes, it keeps the spin from the roll and you get rifled spin.

I don't throw with consistent ammo, so I don't know how I'm throwing honestly.

I don't know how effective fins would be for lead ammo, the point of lead ammo is to fly farther, and the fins only add a lot more resistance.

It's counter intuitive. Low density/light weight projectiles fly further with find because the fine help stabilize them. High density projectiles fly worse with fins because they don't affect flight as much but create the same amount of drag (or more) as if they were on the lighter projectile. Sling projectiles tend to spin a lot as well, so a lead nerf football would need helical fletching to not prevent the projectile from spinning and the helical fletch would create even more drag. And the fine would need to be really big and paddle shaped to prevent tumble, now that I think about it.

The best idea I've come up with for self stabilizing amo is an egg shape with a hollow cavity in the small end to lower the density on that side. It would stabilize like a badminton birdie without suffering from too much drag once it does stabilize (hopefully). Or you could just learn to sling biconicals the right way...

The way I always understood it, fins always = drag. It's a way of using drag to effect directional flight.

I believe that any reduction of density for a projectile of a constant mass will result in an increase in volume/size.  As long as your section or diameter (think head-on view) does not increase, you should gain stability in flight.

First the ammunition needs to be symmetrical at least in two planes.Round cross section so it doesn't tumble,vibrate in air,also it can roll out of the pouch smoothly.The construction of the pouch also can help or ruin this.
A balearic type split pouch is ideal for biconicals.
Then,the slinger should do his job to deliver a rifle spin.
The best probably an overhand with the palm, facing the target,CORDS ARE PARALLEL,not twisted in any way.
The projectile placed precisely across the pouch.
The more elongated the projectile,the harder it  is to rifle spin it.
2 times as long as wide is easy to rifle spin.
3 times as long as wide is tricky.
4 times as long as wide is quite challenging.
I like the 2.5 times as long as wide formula. Many of the ancient ones are like that too.
This applies to every kind of material made ammo.
Doesn't worth attaching fins IMO. i've cast lead ammo with fins,made clay ammo with fins. It may help with a fooled rifle spin,but causes huge drag usually. The elongated football shape is  so efficient,it just has to be launched properly.
No need for different density layers and other wonders.
A nicely designed and cast lead that's slung properly will fly like a bullet.