Sounds like the headphone jack is messed up. Usually the tablet can sense when headphones are plugged in and shuts off the speakers. Sounds like yours thinks the headphones are plugged in until you actually do plug them in and then it thinks they're unplugged! It's also possible something is wrong in the operating system but I think that's unlikely.
I've sometimes been able to get headphone jacks working by cleaning crud out of them, so that might be worth a try. If that doesn't work it probably needs to be replaced.