Septic odors stink!!! The rotten egg smell could have a couple sources. If from the septic system, look for a floor drain somewhere in the area. It will have a P trap just like under the bathroom sink and when they sit for a long time with no water draining, they can dry up. The water in the trap keeps the septic oders from coming back up into the drains. If you have one, pour a gallon of water in it. Then do it once a month or so. Also look around in the general area for other drains. Washing machine drains and laundry tubs have traps that can dry up. Check for an unused basement bathroom drain.
Maybe there is a hole in a drain pipe. Was any new construction done in the area? Sometimes a nail may get hammered into a drain pipe behind a wall.
It also could be the roof vent is partially plugged (which causes traps to be sucked dry when toilets are flushed), not high enough on the roof, or sometimes weather conditions cause the vented smell from the roof to come down to the ground. The sulfur smell (H2S gas) is heavier than air and will settle.
I doubt that it is the softner. If you are getting the smell from running tap water, then the water in some places will release H2S gas when heated. There are generally two fixes. Replace the magnesium anode rod in your water heater to an aluminum one or make sure the temperature is cranked up as high as it will go. Was a new water heater put in recently?
If it was the septic field you would probably notice the smell out there but like Regina said, check the tank and every few years or so, they need to be pumped out.
Post back if you find a floor or other drain or got a new water heater.