Red light won't turn off after charging Galaxy S3

Hello!  I purchased my first smartphone, brand new S3, in April.  Everything has been great until a week ago.  After charging, the red light would not turn off even thought it was fully charged. Then the battery began to drain significantly faster, at a rate of 10% an hour, when it had been holding a charge for up to 3 days, depending on use. Even if the phone is completely turned off, it discharges at the same rate.  There have been no new downloads, additions, etc., to the phone, as it is used mainly for calls and texts, with the occasional internet use. No games or things like that.

Any and all, help, suggestion and tips would be greatly appreciated.  I do want to caution that I am not tech savvy when it comes to this phone.  Samsung has offered to replace the phone, but before doing that, I wanted to explore other options, if there are any.

Thank you!