Don't go for a new headset. What you have is the best threaded headset ever. Replace or fix the cartridges.

The #1 killer of headsets is false brinelling. The brilliant Shimano design largely eliminates this, but is not totally immune. There is a chance that your cartridges also suffered from water intrusion, and that you may find a corroded mess on the insides. Either way, you should to take them apart and then report back.

With false brinelling, the balls dig little 'pits' into the races. This is what causes 'indexed steering'. With almost all headsets, you can eliminate the indexing by putting more ball bearings into the headset that were there before. This way, the balls now longer have the chance to line up with the pits in the races. This sounds like a kludge solution, but the headset is restored to silky smooth operation, and buys you another 5+ years of headset life.

In your case, you may as well replace all of the balls. Make sure you use the same size balls as were in there before - probably 5/32". This should only cost a couple of bucks. Crack open the carts with a small screwdriver, pull out and discard the old balls and the plastic bearing retainer, and fill up the unit with new balls. That is tight with balls, and then subtract one.

If for some reason the cartridges are really really in bad shape, trust me: try and get new cartridges. Replacing the headset with something else will involve possible stack height incompatibility, and as with any pressed-in headset install, the ever-present risk of messing up your head tube or the fork. And again: there is no 'upgrade'. Any new headset you get will be inferior to what you already have; I've owned all of them.