This is what showed up in the Message Logs whenever I tried to send the message: "(Error: 30007) Message filtered Your message content was flagged as going against carrier guidelines." According to this guide:
https://support.twilio.com/hc/en-us/articles/223181848-How-Does-Message-Filtering-Work- , if you collect 3 or more examples of Message SIDs that have the “undelivered” status with error 30007, and then contact the Twilio Support team, they can review your messaging and determine if an error was made. Thanks for the help guys!
EDIT -- Twilio support got back to me, and apparently my messages were getting filtered (and thus getting the error 30007) because my carrier for my personal phone number is Sprint, and according to Twilio support, messages to T-Mobile (including Sprint) phone numbers may get filtered because they have not launched their A2P 10DLC solution. Other carriers like Verizon and AT&T have already launched their A2P 10DLC solutions, so this issue does not occur if the carrier of your personal phone number is Verizon or AT&T. But again, this issue might occur if your carrier is T-Mobile or Sprint.