Yep, the others have already nailed it, but you can surely use that gun. You just have to find the range limitation and stick to it.
I'm guessing you're just starting out turkey hunting? If so, use the gun you have for a while, and I bet after a season or two, you'll have some pretty specific ideas of what you want to use as far as gear and such.
I started out turkey hunting 5 years ago with an 870 that my dad gave me when he was thinning out his guns. Despite looking at several purpose-made turkey guns and thinking "I love this gun!" I am still using my 870. I've added some stuff to it to make it more to my liking for turkeys, but it's the same 870 he shot ducks with when I was too young to hunt.
Since shells can be expensive and you're shooting a modified 2 3/4" gun, if it were me, I'd look at copper-plated lead shot and avoid the HeviShot. You're not likely going to be shooting to or beyond 40 yards, so I'm not sure if the HeviShot will be worth the price you'll pay to get it. If it were me, I'd be starting with whatever copper-plated #5 loads I could find that were cost-effective and see what they did, then go from there.
I've usually looked to see at least 90-100 pellets in an 10" circle around my point of aim. Once the count gets lower than that, I consider it to be too far. With my current gun, 45 yards is still good, 50 is really pushing it....but that's running 3" HeviShot 6's, 2oz loads, with a Jellyhead turkey choke.
And to be honest, the three shots I've gotten in my time turkey hunting have all been within 35 yards.