Being in Florida and on City Water - and only the cold water is affected.
Your water heater is not the culprit.
However, you stated a very important fact and left off another possibility - you are in Florida and many homes in Florida might be equipped with an irrigation system, and if your home is equipped with one, see the following.
Usually, the irrigation system will have a back flow valve (check valve) to prevent back flow from the irrigation system running into your home.
There are also water filtration systems for homes, does your home have a filtration system ?
The other possibility is that you have a cracked pipe running into the house or a tree root has found your inlet water line and broke it a bit. Turn the water off inside the house for a 1/2 day (Take a picture) - go out for a while and when you come back take another picture and see if your water meter budged even a blip.
Your hot water heater is cooking the water - so the smell is not there - but the sulphur is sticking to / burning on the tank heating rods.