i have used these in college and worked many dell d620 laptop and i can tell you from past experence its the battery.. i hope you have already replaced it with a new one..
i purchased an aftermarket type and it lasted serveral hours; for the money it was a good buy...
don't leave your laptop connected to the charger for extended periods of time....
the dell d620 are old but still durable workhorses; espiecially for the telemarketers.. its slim, got lots of power nice wide screen and can take a beating..... parts are easy to find and very affordable.. Dell licked the spoon with this model and called it good... the rest is history...
enjoy.....