Okay...but won't a new rod eventually cause the same problem unless you address the source of the problem? Plus the OP states the cold water supply, which never passes through the hot water heater, has the same problem (although to a lesser degree).
Is it possible that water from the hot water heater is leaching back into the cold water supply line?
I'm certainly no expert on the matter, never had the problem, but logic says that he has a supply issue first and foremost, and the heater element is simply accenting it. Of course, after addressing the supply side and eliminating the odor source he may have to replace the rod to get rid of any residual odor.
My approach to problems like this is to try the easy things that cost little money first.
Draining and flushing the HWH costs him nothing but a little time. If it doesn't work, then add some peroxide to the HWH also costs next to nothing. If neither of those work, then maybe it is a supply problem, and a filter would help.
Kind of like "did you check to make sure it's plugged in?" question when you call IT support....
I don't know about that, but I'm reasonably sure it's not leaching back 12 miles down the pipes to also cause the same smell/taste at the medical office she works at. No one else at said office detects a problem.
Also, the smell is starting to be less bothersome to her after a week and a half, so coupling that with the fact that no one else at the office thinks it smells weird, I'm thinking it's an acclimation issue. Once you're there for awhile, you don't notice it. Also, I've always thought municipal water in coastal areas tasted terrible (enough so to bring my own water to get me through the duration of a trip), so it's not like this is a smell/taste that's unique to this particular house.
Assuming that's the case, that puts us either just getting used to it, or putting on a filter.
I'll gladly give it a shot giving the water heater a flush next time I'm at the house (can't hurt if even just for the possibility of squeaking a little extra life out of the unit), and trying some peroxide, but I'm still thinking I'm needing a filter.
As stated, could be the anode in the water heater. The cold could smell like it because it's going through the same fixture. You could try turning the water heater all the way up for a few days after flushing and see if it cooks out. Or pull the anode completely out. Municipal water shouldn't smell like that even if it is from a well. A whole house system wouldn't hurt. A 20 micron pleated filter before a 5 micron carbon filter would work well.
100% Authority on the above post^. Replace the anode rod in the heater, flush it in any way you can. There's a million recipes for putting bleach in the anode rod hole while youre doing it but I warn you it take longer than you think to flush the bleach out through the plumbing. Crank up the T-stats, there's a heat range that causes that bacteria but the high end of your heaters capability will actually prevent it. If you're any where near sea level (and you are) a whole house filter is a basic must- smell or no smell. The clear plastic housing ones at the retailers, (the size of a 2 liter coke bottle) will do fine. Start off with the white sediment style cartridges for a month or two, it will show you what you're up against. After a while move over to the low micron (black, charcoal) cartridges, theyll do the same cleaning plus clear up the taste and odor.
Anode rod doesnt cause the rotten egg bacteria, it's job is to help prevent it. Stagnant 100° water in the tank that can maintain 150° causes it. It's a known thing. People aren't fixated on the water heater: plumbers are, because they've solved this problem 100% on many many service calls in the ways stated above. It's not even an unusual service call, it's a routine deal
I thought the water heater anode was a sacrificial rod like ships and submarines have. Used so the water eats them up instead of the tank or ship hull.
Some rods like the magnesium ones do cause the sulfur smell. I'm not sure why. You could remove it or replace it with a different type. Probably a cheap thing to try first.
I doubt that's the OP's problem as he says it's in both the hot and cold and where his wife works. Lots of places and wells in the South have sulfur water. You can really smell it when someone takes a shower. My dad always like the taste of sulfur water.
It takes some acclimation like living near a paper mill.
Limiting Hydrogen Sulfide in Water
Unfortunately, the City of Douglas cannot eliminate the hydrogen sulfide (and its unpleasant rotten egg smell) from the groundwater deep wells. However, we are taking steps to limit its effect by initiating an intense flushing program, as it tends to build concentrations up on dead-end lines. We have also increased our disinfection process to help mask the sulfur smell.