Welcome to iMore.
You can try a hard restart on your iPhone by holding the Home & Power button at the same time till you see the Apple logo. Make sure your iPhone software is updated. You could try turn Background App Refresh off. You would fin it at Settings>General>Background App Refresh. If these do not help then you may have to restore it to fresh and start over with it.
As for getting a new phone, unless there is a hardware problem, Apple would most likely restore it and if that solved the problem, they would send you on your way. If it did not solve the problem, you most likely would get a new phone. That is Apple's call though and ultimately up to them.