Soft water is probably going to be a big problem, due to the sodium content. This may be why your plants don't appear to be absorbing water well (and why they're so prone to sunscald). Potassium deficiency is another likely cause of the thin leaves, if you're not using potassium to soften your water (and definitely of the weak look to the plants). Potassium strengthens plants a lot and helps them absorb water (sodium can interfere with water absorption), although potassium in over-abundance may cause stunting and yellowing, too. However, if it has both sodium toxicity and potassium deficiency, I'm not sure that potassium will help in an already salty soil. I don't know, though, but people say it's a bad idea to add fertilizer and mineral salts to extra-salty soil.
The light-colored patches appear to be sunscald. A nutrient-deprived plant can seem to be more prone to stuff like that. It's possible that there are too much aridity and heat stress for the plants, too. Potassium also helps with heat-tolerance (as does magnesium). The plants may also have had low light levels before transitioning to the hardening off phase. Extra organic matter may help.
Misting your plants (not with soft water) may help their leaves to absorb more water, and may strengthen the plants, due to the wind-like pressure of the water, and due to the foliar application of whatever nutrients happen to be in the water.
Anyway, the plants look like they need extra nutrients and to adjust to the sun, but it sounds like soft water may be the main problem. Rainwater is really great for plants, but regular water (or even city water) should be an improvement over soft water. If you can give the plants new non-salty soil and turn the softener off when you water, that may help (unless the regular water kills plants). City water can be pretty bad sometimes, but it really depends. Our city water works, but plants definitely prefer rainwater or filtered water to our city water.
Too much salt may impact the roots of your plants. I had a pepper with too much salt that died of root rot. This is likely because it didn't absorb enough water (so, it got too much water in the container; you'd think salt would deter the microbes that cause root rot, but it didn't in this case). So, a drying-out plant, in this case, probably would have done better with little water (not more).
If you've been growing them on soft water all along, the salt content may not have begun super high, but may have accumulated. Salt (in the form of sea minerals, at least) can be beneficial to plants in small amounts, but in my experience, there comes a time when it's too much (and that time doesn't necessarily come right away).
I'm not sure how much hope there is for your plants, but I would replace the soil (or else transplant somewhere shady, removing the old salty soil when you do) and add potassium (and non-soft water). I would also mist the plants (without soft water). I would add phosphorus, too, but that's me.
Repotting may stress the plant further, though. I'd probably just transplant it at this point (and shade it), as that probably wouldn't be much different than repotting while hardening off. Don't transplant when it's sunny (I prefer to do it at sunset or when it's cloudy).
Whatever you do, don't water the garden with soft water. It may have some long-lasting consequences.