Bob,
Actually, there are two problems with the water - high hardness and a high level of dissolved ions (Total Dissolved Solids, or TDS). A softener will remove hardness ions and replace them with sodium ions, but the TDS will remain high.
When you wash your MH with hard water, hardness can combine with the detergent to produce soap curd. When the water evaporates, the soap curd remains as well as hardness scale (primarily a form of limestone).
With soft water, there should be no soap curd, but there still will be deposits due to the high TDS after the water evaporates. These deposits are sodium salts, like table salt and baking soda. Unlike hardness scale, sodium deposits dissolve easily in water, but then that water evaporates...
With deionized water, there will be essentially no hardness and no TDS. The problem is that, unlike a softener, there is no practical way for you to regenerate the deionizer resin, so you have to replace it. That gets expensive!
Consider doing what car wash operations do. Use a softener for the washing process, which uses the most water. Then use a deionizer for a final rinse.