So I've figured out how to fit text into a shape by using envelope distortion, but when I do it, the text looks weird like in the picture below. How do I make the lowercase letters fill it in evenly and make it so the letters don't slant so much? Is it the font? Thanks!
To the best of my knowledge, nothing which "distorts" a text object is going to work for you.
The primary issue is that distortions see the bounding box of a text object as the "shape" to distort. They do not factor in upper/lowercase type. So the Envelope Distort doesn't know there are lower case, smaller, letters on the right side. It only sees the rectangular bounding box of the entire type object. You can try and alter the mesh created by the envelope to "lift" the right side middle a bit. This might distort those glyphs to a more readable state. In many instances that's more a lesson in frustration than anything else.
You can try outlining the type (
Type > Create Outlines) and then applying an Envelope Distort via top object. This will allow the envelope to actually see the shapes of the type and apply that distortion to the top object. However, in many instances this will result in far too much overall distortion of the glyphs themselves and make readability a challenge in many instances.
Really, for the best possible outcome you need to manually edit each type glyph. That is - create outlines then manually alter the glyphs - change their size, alter curves, move anchors if necessary, etc.
The only way you will get any sort of viable output with this type of distortion, without manual manipulation, is by using all uppercase type which may or may not convey the message in the same manner camel case type would. For a script font, such as the one you are using, all uppercase type tends to look pretty bad.
Note that Type on a Path may also be a viable option for you. It is difficult to state definitively without testing the same font you are using (No clue that that may be). However, Type on a Path won't distort anything other than baseline and x/y of the glyphs. With some monitor size alterations you might find Type on a path yields more visually appealing results.