New toilet handle – not flushing all the way
Toilet tank flapper valve geometry varies all over the place. Some contain air and float when opened, some contain air but have a bleed hole and eventually sink. Some are made to tip over their hinge, counterweighted with water that then runs out through a bleed hole. The exact solution to your problem will depend on which kind of flapper valve you have. 

In every case, though, the trick is to adjust the chain so it is just tight enough to pull the flapper open, and no shorter. Once the flapper has been pulled far enough to stay open, it must be free to continue moving on its own. 

Your description suggests that your chain is too short. That's why @Jimmy Fix-it wants you to experiment with the chain -- not to see if the flapper is working properly, but to see exactly how far you need to pull to trip the flapper's action. 

The other thing you need to do is to inspect and adjust the angle of the chain. At the point where it becomes taut, the top of the chain must be directly over the flapper or its hinge. Pulling at a large angle may not open the flapper far enough. 

Trouble with toilet flush plumbing is very common. I am always surprised when a flush toilet in my house works properly for as long as five years without some kind of attention. Just be thankful your problem does not involve a leak.