DoubleVision

I have a Compaq Presario 3000R laptop. Recently I noticed a problem with the computer running down to hibernate mode (no power) from a full charge after about 1/2 hour to 45 minutes of constant use, just general surfing of internet. Then when I plug in to the powersource, the charge icon comes and goes and vascilates between the charge icon and the battery icon...obviously not continuously charging. To me, this seems like an improper connection. I ordered a new powersource from E-bay and tried that and have the same problem. Is this possibly a battery problem instead? I also want to know if there is any detriment to having the laptop continuously plugged in? Can that wreck the battery? My husband says I should let the battery run down before plugging back in. I feel that until I solve the problem with the intermittent charging, I should just leave it plugged in. I appreciate your sage advice. Thank you.