Cracked egg leaves egg white residue. Should I worry about the other eggs?

Sure. Wash them off with a little cold water, rub them dry carefully with some paper towels in case you want to keep them for longer and not have sticky old egg whites on the outside.

Edit after the question was substantially changed. This answer wasn't initially concerned with the fate of the broken egg itself, but the others around it.
If the white 'exploded' bin it. If it 'leaked' then that egg is compromised. If you broke it, eat it today. If you don't know when it was broken, discard & treat the rest of this answer as it stood before you changed the question.

Don't wash European eggs.

US eggs already have had their natural protective layer washed off, so this action would be sensible to prevent further contamination.

EU eggs still have their natural protection, so should be left alone. Wash right before use if you're worried.

I am new to your site so I may not understand how to view all the answers or the entire discussion, but I did not see the solution my mother taught me. You can use the same "trick" to test for freshness that you can use to test for potential contamination in your egg, so I will give the answers to both. (I noticed some people do not like you deviating from the original question but, hey, I'm old. I figure it is NEVER too late to learn something that may come in handy some day.)

Place your egg in a cup or bowl of cold water - cold tap water, not cold as in refrigerated.

Make sure there is enough water that the egg can stand up on end in the container and still be covered with water.

If the submerged egg lies flat along the bottom, the egg is very fresh. ENJOY! If it doesn't but was sold as fresh, you might want to mention that to the vendor.

If the egg stands up on one end it is not so fresh, up to maybe a few weeks, but still good to eat.

If the egg floats, it contains air and should be tossed. It is not only old, but air can contain contaminants and air is how contaminants get into your egg.

Believe it or not, I have had eggs that have been in the refrigerator for months that stand on end in water and are just fine. But remember ALL pre-washed eggs can contain air and contaminants that can cause Salmonella-and-such so always thoroughly cook your eggs. If you must drop a raw egg in a drink, you might want to make sure it lays flat on the bottom of a glass of water first.