Goto the verizon store and get a new battery.
I had an issue like this recently where all olf sudden my phone would't make it thru the day on MY standard useage. Previous, it would go between 24-36 hours, doing nothing different, it wouldn't make 12. I have a droid from last Nov.
I deleted some apps. I did battery pulls. I tried everything for about a week. No difference.
Called VZW tech support. Went thru the usual stuff, they said they couldn't authorize anything over the phone, have to goto store.
Went to store. Wanted to test phone for an hour. Like I have an hour to sit around the vzw store after waiting 30 minutes to speak to tech service! Convinced them to give me new battery.
Problem solved. Phone lasts about 30 or so hours with new battery and my useage.
Li-Ion batteries have a fixed amount of charge cycles they can go thru, then they don't hold charge well.
Oh, and if your phone is really turned off, then there is very minimal battery drain. You should be able to leave it off for a week, and when you turn it on, maybe it will be down 20-30%. However, your phone circuitry could be bad such that when you turn it off, it is really not turning off and going into a down state. So it might be the phone that is bad, and not the battery.