Keep Dropping Wee on Downhill Stops

(Preface: I am 5'-10" on a Wee with a touring gel seat).

Work has me temporarily relocated from Chicago to Chattanooga, TN and now I have elevation changes, mountains and twisty roads! :thumbup:
However this also caused me to let my Wee take some naps on the road:

Nap #1: I am hitting up the twisty mountain roads with a riding buddy. He takes a turn too fast and dumps his bike in a downhill hairpin. He was down in the roadway so I came to a stop in the middle of the curved, banked road and tried to put my feet down, lost balance and fell over.

Nap #2: A bunch of us rented a chalet in the Smoky Mountains one weekend and to get from there to anywhere, you have to navigate a maze of twisty roads with large downhill sections. I came to one intersection that had a stop sign at the steep downhill part that also had a curve to it. Once again, came to a stop and felt the unbalanced weight of the bike and let it fall over.

What am I doing wrong?