What am I doing wrong soft boiling my eggs?

If the yolk is settling to one side (typically the wide, bottom end) try placing the carton on its side in the fridge the day before you use them. This will move the yolk back towards the centre.

Also, ensure your eggs are cold before and start them in boiling water. This will allow the whites to set much quicker than the yolks, preventing the yolks from migrating while they cook. Cook in a covered saucepan with about 1/2 inch of water for about 6 1/2 minutes. As you have been doing, immediately cool in an ice bath or under cold running water.

Finally, depending on your location, equipment, and how set you want your eggs, you may have to adjust your cooking time. Try taking the eggs out 30 seconds apart and see what provides the best time for the results you want (a digital timer will help here), then stick with that in the future.

Cook's Illustrated/America's Test Kitchen has a pretty foolproof method.

The trick is exactly controlling the cooking time and temperature. When you have a large pot of water, if you add in eggs, the temperature of the water drops, and has to heat back up to boiling. This is a highly variable process, depending on volume of water, heat of the burner, heat capacity of the pot, number of eggs, etc.

What they've arrived at, instead, is to just use about 1/2 inch of water in the pan. You heat it up to boiling, and when the eggs are added, there's so little water in the pan that it gets heated back up to boiling very rapidly. This makes the temperature and time cooked very, very consistent which allows for consistent results, even when varying between 2 to 6 eggs.

They are actually steaming the eggs, essentially.

While their site is behind a paywall, many others have shared the method on other sites -

Bring 1/2 inch water to boil in medium saucepan over medium-high heat. Using tongs, gently place eggs in boiling water (eggs will not be submerged). Cover saucepan and cook eggs for 6 1/2 minutes.

The Bitten Word: Foolproof Soft Boiled Eggs

NOTE: Cook's Illustrated recommends a shorter steaming time, if I recall correctly, but that yields a very, very runny egg and whites. I think this site increased the time for more solid whites, and soft or runny yolks. You'll need to tweak it a bit between 5 to 6.5 minutes to find how you like it. The nice part is, once you do, you can easily replicate it!

I've never had any luck making soft boiled eggs using the heat and stand method, hard boiled that's my favorite method. I think when you take the pan off the heat you get less convection, which would account for the uneven cooking, but that's my own theory which I haven't backed up with any scientific process.

For soft boiled eggs I would suggest putting cold eggs into cold water, then bringing the water to a soft boil for exactly 5 minutes and 20 seconds (this is for UK Large eggs which would equate to US XL), then immediately spooning them into ice water. I use this technique often and it gives me very consistent results, the white is hard and the yolk is still gooey. You will have to vary the time depending on the egg size you use and the hardness of the yolk you want.

Use a skillet with a lid. Bring unsalted water to a boil. Lower heat to a simmer. Add 3 - 4 drops of white vinegar. Crack the egg in its own small cup. Lower the egg into the water. (Repeat if want more) immediately put a cover on the skillet, turn off the heat. Let it sit exactly 4 minutes. Scoop out with a straining spoon. Serve! (don't put them in ice water!)