Softeners will not remove H2S from water. If the smell is mostly from the hot water, there are some fairly simple things to do to help. There are devices to remove H2S coming in with the water, but those are typicall going to look similar to another softener.
Tell us about the air in the pipes? Is it possible this only occurs when the softener regenerates? Keep track of that. If that is your symptom, easy solution.
A smell when using the washing machine could be a drain pipe problem.
How long since you had the septic tank pumped?
Thank you so much for taking the time to reply!
The replacement of the water softener was actually a completely separate issue. The brine tank kept overflowing and it was old, so we had it replaced. The water in the lines issue had been going on for about 9 months and I was naively hoping that would solve the issue, but it did not. When we had the plumber out to replace it, we also wanted him to help us with the air in the lines issue, but he pretty much did not want to be bothered. He said that if replacing the water softener didn't help, we should just call a well guy. Maybe we need to, but I feel like I really want to know that we've ruled out any other issue first.
The air in the pipes is every time I use the kitchen faucet; not quite as bad with the bathroom faucet; not noticeable with the shower faucet (I think because we are using hot water) and sometimes very noticeable with the toilet tank filling back up after flushing.
The smell is a sewer like smell and we smell it in the main floor bathroom and only when the washing machine is running (which is in the basement, essentially right below the bathroom).
Our septic tank is a holding tank, so it has to get pumped regularly (probably every 4-6 weeks).
I'm sure this is a really stupid question, but if we have a drain pipe issue causing the smell, could a drain pipe problem also cause air in the lines?