I have searched the forums in search of a solution for why my race widget does not update, and the only solution I found was to clear my browser cache. This did not work, neither in Safari nor in Firefox. BUT I did find a solution on my own:
I simply clicked on the race that the widget was stuck on which took me to the race's DM page. At the upper right hand corner is a button that says 'remove race'. I clicked on this to remove the race and my widget instantly updated to show my upcoming race. I know that it is not ideal to lose your race history, but it's the only solution I can find to get the widget to work.