Suddenly
No Punting
Also, does the Innocence Project tend to only take on cases where they are convinced the person convicted was actually innocent or do they tend to take on cases where there were some procedural errors?
Generally the former, although "convinced" really isn't a thing. Suspect or have reason to believe is closer. They did and probably still do ask their clients to pinky-swear their innocence. Which strikes me as adorable naivete, but they are doing the Lord's work so no judgment there.
There is also that post conviction law is a mess so every case is going to be a combination of these two. Often you have legal error and you are looking to show factual prejudice, or you have great factual evidence and are looking for a way to get it to matter to a court. Seldom both. Usually neither.