Welcome Turtlepee,
If you want to or have to use pure cotton then a braided sling is probbly the best way to go. BUT, they are fairly prone to slippage and for that reason may not be the ideal soil for your budding obsession. May I suggest what is referred to on this site as a "drawstring pouch"?
Denim is good, and cotton t-shirts WILL work, but fold your pouch into layers, and sew them together. This will help avoid excessive "cuppage" of the stone; your 50' throws were probably the result of this little teaser.
Weight is not much of an issue but sling length and material flexibility are, so experiment, be prepared for a steep learning curve.
Go now, frolic and make merry in the glory that is "sling"