It isn't elegant, but the handiest storage option I have found for breakdown paddles is those tall plastic oval-based totes with the round perforations and pair of cut-out top handles (sorry, no link to a photo -- though I see them in countless stores, from K-Mart to Walgreens, no images turned up in my various google searches). I call them "Swiss cheese totes". I have several boats and sets of outfitting for them that I often share with friends, so I fill each bag with the two halves of a paddle, a PFD, a bilge pump and spray skirt (and, for my personal gear tote, also a pair of paddle gloves and a deck bag).
I also keep one filled with my cold weather clothing kit: neoprene dive booties, cap and gloves and dry suit. Easy to grab and I know everything I need is in there. And there is the whitewater kit with neoprene skirt, float bags, helmet and WW paddle
The paddles, tucked between the skirt and PFD, are well protected. everything can drain and dry easily and the totes are rigid enough to stand up in the hatch of an SUV or wagon, tuck behind the front seats or the vehicle floor or lie down in a trunk. Between trips they can be hung on a large hook on the basement or garage wall. The bottoms are solid, so when stuff is in the car after an outing, drippage doesn't get on the carpet. You can set the thing in the yard and hose everything off right in the tote, handy for salt water or mucky pond outings. Then just hang it to drain and dry.
I tried sewing polarfleece and then neoprene "socks" for my paddles, but found them a nuisance. The totes routine is highly convenient, easier to carry and I don't forget anything now when I get the urge for a spur of the moment paddling outing. It also makes it easier to inventory all the gear when you get back in the car -- every paddler returns their kit to the bag that they brought it in. The totes are easily labeled with a Sharpie.
Since I've been using this system (about 3 years) I have not had one incidence of getting to the put in and realizing I was missing an important bit of gear. Sure saves time rooting through tubs of stuff when loading for a trip.