Check in the battery settings of your phone and you should be able to see if any apps are draining your battery. It should at least let you see if there are new apps you installed that are vampires.
Believe it or not, batteries physically change when they "go bad." You can remove the battery from your phone and place it on a flat surface. If it is significantly degraded, it will probably have a slight bulge. Note that this is not a major, my-battery-is-about-to explode bulge. It can be difficult to see, but while resting on a flat surface (glass is best), flat side of the battery down, try to spin it. If it spins freely, there is a slight bulge that indicates the battery has physically, or more accurately chemically, degraded. If it doesn't, the battery is probably fine.