Sorry if this has been addressed somewhere else, but when a new build is released, am I able to toss my old save file onto it? Or somehow update my existing client to not have to go through the entire thing again to see the new content?
You cannot load old saves in newer versions without a chance of the game breaking, no.
But if you start a new game, and say you've played the game before, you'll get access to the chapter select screen where you'll be able to get back to where you left off immediately. Finding the exact chapter might be a little tricky without asking, but every time a new build is released, the latest chapter is always at the bottom (might need to scroll for it though).