Its hard to say for certain... sounds like a bad battery though. The chargers tend to last forever.

the best way to find out for sure... have the local hobby dealer put the suspect battery on a cycler (or a local club member... someone who is nice and helpful...) The cycler will test and determine if the battery is any good at all.

The charger will get hot if it has an internal short... or if the battery is bad. (either can make you woorry about the charger case melting) The internal short would have to be in a bad spot to have the LED light up still... and if it is shorted n that manner... the LED would be lit even if the TX were not pluggedin for charging.

Be VERY careful with plugging and unplugging the battery from that radio. Note where the red wire is and mark it. some people have been destroying thier TX's by plugging JR batteries in backwards.