FBinNY 

You definitely have headset play. That's not in doubt. There are two scenarios/

Play within the bearing, which would adjust out if the headset is tightened by way of the top cap (stem loosened, of course). If you've adjusted the headset to where it's noticeably stiff and there's still play, then you need to consider the other option, play between the headset and frame or fork.

Drop the fork and check that the crown race is tight (immovable), next check that both upper and lower frame cups are immovably tight. These three are press fits, so if you can move them that's likely your problem. If you have a TiG welded frame I suspect that you may find the lower cup loose because of bellmouthing of the non-reinforces lower headtube.

If the three cups are solidly tight, it gets complicated. I've seen headsets where the steel race is loose in the alloy cup, and other rare similar anomolies, so you have to run through them one step at a time.

BTW- if you have an integrated headset some of these have been known to have wear on the conical interfaces which make zero paly adjustment impossible. This is fixed by re-machining the interface a hair deeper to re-establish the correct fit. If that turns out to be the issue, you can prevent re-occurance by bedding the races in a bit of adhesive to eliminate creep under load.