I finally bit the bullet and subscribed to Amazon's Music Storage. I purchase the majority of my music from Amazon, use their app on my Nexus 5, and occasionally use their web cloud player at work. By default with Amazon Prime, all of my music purchases are stored in the cloud for free, but paying another $25 a year allows me to import up to 250,000 songs.
I only have roughly 9,000 MP3s, but being able to access that content from essentially anywhere for $25 a year is definitely worth it. Unfortunately, it doesn't seem that you're able to use Amazon Gift Cards to pay for the service -- I guess because it's a "subscription" they want you to pay for year-to-year.
Importing the music took a few hours. Their import tool attempts to find a match for each of your songs in their music library (assuming it only checks by ID3 tags), and then the remainder which can't be found are uploaded. It definitely took a few hours to complete the import.
Now begins the long process of sorting through the music and trying to create playlists of the stuff I actually want to listen to. I'm averaging about one letter a day, so I'll probably be done organizing closer to June.