Rule of thumb for the locknut is that it must have a minimum of three full turns of engagement. It sounds like you are just shy of that. Rather than replacing the headset, the most cost effective option is to search out a thinner lockwasher. A lockwasher that is one millimetre thinner than your current lockwasher should gain you about an extra full turn. The most drastic solution would be to mill a half millimetre off of the top and bottom of your headtube.

Good luck in finding a headset with a smaller stack height. My experience with the relatively modern Shimano headsets is that they have the smallest stack heights. This crafty engineering trick pretty much forces the owner to use Shimano when it's time for replacement. It also means that a Shimano headset can usually be fitted as a replacement for any other headset by shortening the steerer tube, which is a rerlatively simple operation. Once you've done that, your are tied to Shimano. They pretty much tied tied up the aftermarket headsets for a while with this approach. Hopefully, the other manufacturers have caught on and matched their stack heights.