If I edit my widget while a giveaway is running, will I have to re-embed the widget code?

Nope! If you edit your giveaway (entry options, prizes, start and end times) as it's still running, once you've made the changes you won't have to re-copy and paste. The changes will be reflected on your widget(s) that are on your blog / Facebook pages automatically.

However, if you edit the look of your widget through the Rafflecopter DesignLab and want to use a new theme, you'll have to re-embed the widget with the new Rafflecopter code that uses your newly designed theme.