I'm in the same situation as you. I guess any of the websites where I usually watch the NFL games the day after (nflfullhd, fullmatchtv, nflreplays...) will upload the Draft as well. If not I will search in Google: 'NFL Draft 2017 replay' or something like that and pray to don't see any spoiler.
BTW, who y'all got for your favourite team with their first round pick? I'm praying for Adams/Hooker/Foster falling at 17 for my Redskins.