Eggs should be fine for a week or 3 after the date. They are the ONE food that I do not follow the date closely. I have used eggs up to a month after the date if it is for a cake or something.
While the float test can help you find the older eggs, I am not sure what it says about their use. The float test tells you how much AIR is in the egg.
If you washed the eggs with the shell on, use them ASAP. Eggs naturally have a waxy coating, one we don't notice. This protects germs, etc from getting into the egg. that is also part of the purpose of that thin membrane that is around the egg.
Why do hard boiled eggs sometimes peel easily and other times put your nails in danger ofthose tiny bits of shell?
The egg has that waxy coating and the inner membrane to keep air out. But there is a tiny amount of air inside the shell between that membrane and the shell. As time goes by, air seeps in through that waxy coating and the inner membrane.
When you peel a brand new egg there is that small area that has a crater, for lack of a better word. It is bigger in the older eggs. This air pocket is what makes the shells come off easier. It gets all around the egg as the egg ages, making it much easier to peel.
We did a test in a college food lab. We boiled 3 pans of eggs and then peeled them. The first one was brand new eggs from the University chickens (we are an ag school, so there truly ARE University chickens!). The next batch was 2 weeks older, the 3rd pan was 4 weeks older (all eggs were delivered to our lab on the day we started the experiment.).
One group used no special stirring, no vinegar, no salt.
One group stirred halfway through the cooking period. One group used 1 tbsp of vinegar, one group used 1 tsp salt.
The ONLY thing that was consistent was how the eggs peeled at each stage. None of the test groups was any different from the control (the group that was just boiled and cooled).
We DID find that if you stir clockwise and then counter clockwise about 3 or 4 times during cooking that the yolks will be in the center rather than pushing the egg white so thin on the sides. It is better to stir early in the game, as the whites are setting up and thickening - this will make it so the yolk can't push to the sides.
A helpful tip for devilled eggs, in my opinion.
I wouldn't worry about the date if you got them from a store you know well.