The float test has nothing to do with quality or safety, only with the age of the egg (the air cell in the age will expand as the egg ages as the shell is porous).
When eggs in the US are processed the protective coating on the shell is removed, so they need to be refrigerated. Only about 1/10,000 eggs has salmonella. While I won't, on principal, advise whether your eggs are safe to eat, I will say that I believe the risk is low based on what you have told us.
The risk of spoiled food has many factors. Temperature is one. Time is another. Origin and processing are others. You should always keep your eggs in the back of the fridge where it's colder. Keeping them in the door (where the egg keepers are often built in) is the worst place because you open and close the door all the time so temp goes up and down, contributing to additional risk/premature aging.
If you are worried, using the egg in a way that cooks it through will reduce your risk (baking, scrambled medium or hard—yech, hard cooked (aka hard boiled)).
You can also pasteurize your eggs using sous vide which allows them to be used safely for many recipes where the yolks are not fully cooked. This might be your best bet if you are concerned. If you don't have a sous vide device (and every kitchen should!) there are alternate methods that will work well. In the case of eggs, since you only need them to be 135°F for about 75 minutes, you may be able to use water from your tap put into a beer cooler. If your tap water is only 120° (depending on your water heater setting), just boil some to bring up the temp of the rest. Check with an instant-read thermometer and I'd suggest a little above 135 so you maintain that temp for 75 minutes. You can check the temp after 30 minutes and add a little more boiling water if it's starting to dip a little too low. Make sure the lid of your container is kept closed. If you have a probe thermometer (which every kitchen should also have), then you can keep it in the water with just the wire feeding through the top with the lid closed.