I think you’ve got dirt trapped in the tiny creases of the toe box of the shoe.
First thing that I would do is get some cedar shoe trees to stretch out the leather.
-(You ought to use shoe trees for high end golf shoes after every time that you play)

Once the shoe is straightened and the leather is stretched, then a little soap and water ought to cure the issue.

Don’t worry about a little extra moisture, as the cedar shoe tree will absorb it.

Once you’ve cleaned the shoe and it’s dried out, a little white polish can certainly help as well.

I have a few pair of those same shoes. Here's a great way to keep them white, even in the creases.

Wash your DryJoys with light soap and water, rinse well and let dry. Then use the stuff pictured here. It works great and from what I recollect, is was originally recommended by FJ. It's more opaque than regular white, watery, thin shoe polish. I use it on all my DryJoys Tour as well as my other FJ leather models. (except the black ones).

I have had at least five pairs of white croc on white over the last 6 or 7 years. (My earlier versions were white lizard on white if memory serves.) I also have them in black croc on black, navy croc on white, as well as a few pairs of MyJoys custom Dryjoys, one of which I posted on another thread here today. At least three or four pairs of my DJT are two or three years old with hundreds of rounds on some of them. They all still look great. I make a habit of wiping each pair down with a wet rag after any round where they get even the slighest bit dirty/muddy. I also will spend a rainy day, here or there, cleaning all my golf shoes more thoroughly and using the stuff pictured. When I have time to do a top notch job, I mask off everything so the white polish only goes where it's supposed to. I'm a bit OCD. Many of you here have already figured that out!

Guess I didn't wait long enough for my first reply (this one) to post from last night. I thought I forgot to actually hit "Submit Reply." So I posted a shorter version today which I'm assuming will show up below at some point. Sorry for the double reply. Bottom line, Golf White works well.

Good advice above. A little dirt in the creases is IMO inevitable. To combat it, I keep a bottle of Fantastik cleaner and a soft towel in my locker and spray and wipe my shoes after almost every round. A soft bristle scrub brush also comes in handy. When my shoes start to show some wear, I relegate them to ‘mudder’ status and use them on rainy days and for ‘dew sweeping!’

Mr Clean magic eraser works well. Then neutral shoe polish.