Well... washing them is not recommended, unless you intend to use them. The dirt & outer skin contribute to longer storage life. Does that variety normally store well? After they have cured, you may want to put them in sawdust until October (I use hamster bedding for long storage).

Washing is definitely not recommended, and neither is early cutting of the green stems. They should dry off completely themselves before being removed. IMO you have "cleaned" them way too well for them to keep very long. I normally want my tops to have fallen and mostly dried out before I lift my multipliers, too. I do not find dirt on the outside husks to be a problem, especially if it is dry. A little soft hand rubbing will remove most of it without removing the husks, which you definitely want to keep.

Braiding is obviously out for you on these, but where you have enough stem it is a good if time consuming choice. Other wise I would suggest that you store the nests of bulbs in tact, which is also obviously out for you on these. If you have others save them for replanting and eat these. Harvest the others after they go dormant.

Multiplier onions are not harmed by letting the stems go completely brown like garlic would be. They then cure pretty naturally by themselves and if kept ventilated and dry after that should last well past replanting time. A relative of my father had a strain that didn't get planted back one spring, but it grew when planted back the next year almost as if no additional time had passed.

If I wanted to save some can I replant in a cool place? Will they root and keep over winter? I know you can plant scallion bottoms from the grocery store and they rejuvenate. We had 19 days of rain in a 23 day period. They turned yellow/brown and fell over. I got worried and anxious about rot.