We have super hard water here that will destroy a DW in short order. We have to use a de-liming cleaner called Lemi-Shine. We cut it half and half with regular Cascade detergent. Does a great job. You live in California so I'd find out if you need to do something similar.
Yes vinegar can be used to remove the musty odor. Also if you do not have vinegar, you can substitute a packet of unsweetened lemonade mix in the soap cup.