Do you have a well?
It is possible that you are not so familiar with rotten eggs, and you are getting a musty smell. But you may indeed be getting the hydrogen sulfide smell of rotten eggs. If so, I suspect that the washer tub is serving to concentrate the smell in your water like a giant brandy snifter. The hot water is likely to smell more. Test that by filling with cold and see if the smell is largely reduced.
There are several ways to combat the smell. The most effective is a backwashing iron+sulfur filter of some sort. Much of the smell from the water heater is caused by a reaction of the sulfur containing compounds with the water heater anode. There are aluminium anodes that reduce this effect, but a powered anode eliminates that effect. Some will totally remove the anode and replace it with a plug. That will reduce the time before the water heater stops leaking. The purpose of the anode is to protect the glass-covered steel where there are little cracks in the glass.
I have a powered anode. I got it ordered just before I decided to put in the iron+sulfur backwashing filter. I think it will protect the water heater better. I have also flushed my water heater, and got a lot of rocks and ultra-fine blackish stuff. The sulfur compound reaction with the anode produces that.
So where to go now ? Depends. If you have iron too, that backwashing filter is great. Otherwise consider a powered anode. It seems expensive. And removing an old anode is not a trivial act. It is hard enough when the WH is new. They overtorque them at the factory. A big 1-1/8 impact socket on a not-small impact wrench is the tool of choice. Cheaper is to consider washing in cold water if that helps. Regardless, if your WH is over 5 years old, I would consider flushing it even if there are no smells. There are techniques.