My galaxy always thinks its plugged in to charge

SOURCE: Does leaving a cell phone charger plugged in to an outlet at home,without having the phone charging in it,drain energy from your home?

Yes. You are right. Here energy is not the criteria. It is always safe to unplug the charger (when not in use) to prevent the damage of a cell phone due to accidental over load of electricity or short circuits. Moreover Chargers do draw power when plugged in, whether or not a device is actually being charged.
If the charger is plugged in someplace easy to access, like right by where you keep your phone, you should unplug it. But if it would be difficult to have to plug it in every time you need to use it, you'll be fine just leaving it plugged in..and.It still uses energy even when not charging. Any electronic device plugged in a socket uses energy whether it is in use or not.

SOURCE: Power Problem Samsung Exclaim

You can try running the battery completely dead and then recharging, but it sounds as if you have done that already. I would recommend purchasing a new battery as it sounds like it is dead.

SOURCE: my battery on my samsung g600 keeps going flat,

send the phone away for a warranty repair thats if its still in the 12 months warranty. sounds like the phone. since your phone is working with the charger in it, that points to the battery but if you bought a new one then it has to be the phone unless you are very unlucky in getting 2 dysfunctional batteries. But the best way around this is to send it away under warranty

SOURCE: phone keeps restarting while charging

Some chargers dont work with this particular model of phone even though they fit. One that works for me is called a I-GO you buy the tip separately from the cord and its a little bit expensive, roughly 40 dollars, but it works lol.

SOURCE: Samsung Alias will not charge.

try with another battery. if it charge you have battery problem. if it doesnt you have motherboard problem with phone