What seems to be happening is the yolk is getting cooked before the loose egg white firms up - that would be the watery stuff you're scooping out. I'm not sure what you might do to prevent this happening without a lot of extra effort, the temperatures at which the various part of the egg changes are different. You might try cooking longer at lower temperatures, that's fairly standard for 'part is overdone if cooked longer, part is not done if cooked less" in dishes. It might be as simple as using a lot more water and turning the heater off, instead of down - kind of cooking in residual heat for a longer time.
I'm not sure how much time or effort you're willing to spend to fix it, though - you might find it is less trouble just to scoop out the watery bit.
If you're willing to put in extra effort, you might look into tomago onsen, or other varieties of slow-cooked eggs - many recipes use sous vide because its easy, but you can get similar results with a slow cooker, oven or stove, rice cooker, or even a drink cooler, depending on what you have on hand. They will take a lot more time and may be troublesome to make (depending on which method you use), which may or may not be offset by being able to prepare more of them at a time. The egg whites in these preparations are cooked, but very soft and smooth, so you should be able to find a texture with a good dippable yolk (and possibly a dippable white, too) without clear, watery albumen on top of your eggs. There's a couple sites which took scientific approaches, with charts and such for how to get the results you want - you might want to look at them and see of it's worth it.
