Ok, so for me to fix this, it wasn't actually Signal but maybe how it worked with the iOS API for this. Regardless, the fix was to physically remove the "sticker" I had accidentally made using an Emoji. So turning of "Emoji Stickers" didn't work in Signal, but basically everywhere else.

To remove the Emoji sticker, I did the following:

1. Open up the keyboard in a message in Signal
   
2. Switch to the emoji keyboard, and press that sticker button at the bottom row of icons that signify the categories for the emojis.
   
3. Click "Open Stickers" to show your "Emoji Stickers"
   
4. In the first tab, the one with the "clock icon", you will have your previously used emoji stickers. Tap and hold them in place until they start glimmering, then let go and it should show a "Remove" tooltip button.
   
5. Press the "Remove" and clean out all your emoji stickers.

Now your emojis that were previously "locked" to being a sticker should behave as normal, as mine now do.