I was able to resolve my issue of people not receiving the pictures and videos I sent via Messages on the mac. I did this by:
1. Deleting all the conversations from the Messages window.
2. Deleting the ~/Library/Messages folder.
3. I also signed out and signed back in to my iMessage account in the Messages preference window.
I may have repeated those steps two or three times. As a disclaimer, I was on doing this while on a online chat with apple support. The support tech was having me check settings on my phone such as whether "MMS Messaging" and "Text Message Forwarding" were enabled in Settings>Messages. He also suggested ensuring that on my phone the "Start New Conversations From" setting, found in Settings>Messages>Send & Receive, matched what Messages on my mac had set in the preferences window.
I'm not sure how the settings on my phone would affect the ability of my computer to be able to send images, or if it even does at all, but this is how I was able to fix my problem. Hopefully this helps all of you out. Good luck!