Sounds like the headset is loose and needs to be adjusted. You might as well just tear it down and clean and regrease the bearings if you feel comfortable doing so. A good way to test the headset is with the wheels on the ground, hold the front brake firmly and rock the bike back and forth. You will feel a knock if it is loose.

When setting the headset, just barely hand snug the bearing race (first large nut that touches the cup), replace the washer (if using a threaded headset) and while using two wrenches, snug the top locknut to set the headset. Use the method above to test for any play. If there is no play lift the bike off the ground and the weight of the handlebars (wheel removed) should allow the steering to move freely. If it is sticky or just too tight, back off the locknut and turn the race nut just a hair or two and start the test over.

Marc