But your new headset will do the same thing if it is not adjusted correctly when you install it. There needs to be some preload on the bearings so that when everything flexes under load that balls still all remain at least lightly in contact with the races. This lets all the balls share the load rather than putting it all onto one ball and one spot in the races.

To set the preload correctly you want to adjust it first until all the play is gone and the fork turns freely. Then tighten a little more so there is some drag. Turn the fork back and forth 15 or 20 times to aid in seating things if anything needs it. The final preload you go with should be such that you need to press against one of the fork legs with about 2 pounds of push with a finger to get it to begin to turn. At this preload setting everything should be smooth as silk when you hold onto and turn the bars.

After riding for a few hours worth lift the front and check the preload again by pushing on a fork leg. It may be a little lighter but if it all loosens up to where it feels like there's no preload drag at all anymore then re-tighten the preload so at least some of it comes back. I'd still aim for needing a 1.5 to 2 lb push on a leg to get the forks to begin to turn.

Overall too much preload will shorten the life of the bearings by a little out at the far end of their life. But too little preload so that the balls lose contact with the races will shorten the life a LOT and it'll show up a lot sooner. So if in doubt you want them to be slightly too tight a preload than too loose.