Just google, and you'll find this is an oft-discussed topic. Opinions vary. All agree rinsing/washing is good. Some suggest go easy on the detergent. Some say, use soap. All say, use a bag/pillow case if you use a washer (although front-loaders don't have agitators around which a rope could wrap). Some use fabric softeners, others warn of unspecified dangers from doing so. All warn about using the dryer.
There you have it, isn't the internet great?
What makes your lines stiff? The usual culprit seems to be salt, but your location suggests otherwise.
I've used a front loader w/o a bag in the past, can't remember whether with detergent. Definitely didn't use fabric softener.
But sometimes, you just look at rope and know it's time for it to move on to less critical uses, or non-uses. 3/8" docking line goes for $0.60 a foot, which should make eventual replacement quite affordable.