You have to remain on the "Pick History" tab during the Draft for DA to keep syncing. At the beginning of each new Round, the Chrome extension will switch back over to "Players" tab, and you have to switch it back. See here for explanation from the dev:
Thank you John...and Scott! Scott is correct. As long as you click on the Picks History tab in the ESPN draft room once per round (the earlier the better) then the picks can be synced. The chrome extension "should" do this automatically at the beginning of each round, but depending on what you are clicking in the ESPN draft room it may not happen 100% of the time. It is pretty easy once you understand what is happening.