I already had the zinc/aluminum rod on the way and swapped that out last night and today there were no issues with the smell of the water. I've read that for some this solves the issue for good, others it doesn't.
Now I know we don't have a really bad hard water issue because dishes look spotless and there's no film on anything in the shower.
What system did you get exactly for the water softener?
I might try one of the filter systems since those seem to help and are a little cheaper. If that doesn't fix it then I'll look at the water softener too.
Thanks everyone for the input, this has been helpful and keep it coming in case someone else has an issue or questions.