Okay, I figured it out. I believe the problem stemmed from my own lack of understanding of OS X in general and Apache/MAMP in particular (I'm new to both!). Specifically, I didn't realize that OS X runs its own Apache server, and that when you install MAMP, it essentially gives you an alternate installation (I think?). I realized this when I tried to start Apache from the terminal using 'sudo apachectl -k start' and it said process was already running. I then did a bit of Googling and understood why.
Then I implemented the solution. Previously, I had everything under User\Sites and I was getting the PHP script back in plain-text. I moved my files to Applications/MAMP/htdocs. Then I ran MAMP PRO, which showed that Apache was running on port 8888. So I navigated to localhost:8888/~username/ and submitted the form, and the PHP script was processed correctly and I got the echo back.
Someone correct me if my understanding is incorrect. But at least my little project works now. :)