The most (in)famous font that's chock-full of character alternates is Zapfino. Whether it'll look naturally hand-written is up for debate -- once you recognize Zapfino, you're going to recognize Zapfino -- but you can get some pretty expressive variants. The downside of recognizing Zapfino is always being permanently disappointed in how few people actually take advantage of those alternates, though.

You'll sometimes find them in other fonts, though; Hoefler Text's italics have a whole bunch of "swash" alternates, for instance, which can be a lot of fun for titles. (Both Zapfino and Hoefler Text are standard fonts on macOS.)