sehuxle raises the right questions !
It sounds you may have a dud battery, but........
Don't charge overnight via PC, it may drain the battery. Get a wall charger and use that , the charge is far better - and leave charging overnight, you cannot "overcharge" it. Don't charge until its down to 1 bar, 2 at most.
Only switch off completely if it is not going to be used for at least 3-4 weeks, simply put it in sleep mode ie standby.
If this doen't improve it try a soft reset, if that doesn't, the battery is almost certainly dud and you need it replacing - Sony do this - I don't know the Warranty situation where you are.
My 650 is OK for at least 3 weeeks with the useage you have - the above routine should improve the performance of the battery.