Tap still running when switched off?

Most likely you are correct. You can replace the washers yourself - turn off the mains water, and turn on the tap to drain any water. To remove the tap and get to the washer - flip off the top disc on the tap, locate the screw, undo and then you should be able to take off the lever. You will then need a spanner to undo the rest. Using a pliers you can then remove the insert. Locate the rubber ring and replace with a suitable replacement.
If you google replace a tap washer or similar you will even find a video guide.

Thomas
May 2009