So I placed a text box to cover up a fillable PDF and here's the result:
Is there any way set the text box above the fillable areas or remove them entirely?
Never overlap visible form fields, because you can't choose which one is on top, and it could randomly change. If you have the option to make only one at a time visible, that can be a solution.
You need to hide or disable (set as read-only) those other fields.
Can you do that in Reader?
You can't set it up in Reader, but once it's set up you can use it there, sure.
Yes, Acrobat can do it.