"...You cannot get a .357 mag to .44 mag levels..." Or even close.
The .357 is considered marginal for deer out of a rifle. The .44 is no big deal out of a rifle, but it'll do nicely for deer. Assuming the deer aren't huge.
"...8 inch groups at 100 yards...." That's the ammo or the sights. You really need to try a box of as many brands as you can to find the ammo your shotgun shoots best. A shotgun isn't any different than a rifle in that regard. Even a smooth bore.
It a 'youth' model M500? Stocks can be changed with little fuss and not a big pile of money. The only difference between a Youth and a regular stock is the length. Easy, cheap fix.