I assume your "script files" define functions.
"I'm making sure everything is running properly before I throw the code into the GUI." Hopefully you have kept the code, which you used for testing. You might need to repeat some tests.
You do not mention GUIDE. I like that. I never use it.
" by nesting all my functions" Are you saying nested functions?
There is a "design pattern" for making gui-applications, which use nested functions. See GUI Examples using Nested Functions. I relied on that pattern for several years. (That was before Matlabs new OOP-support.) However, your "domain-specific" functions should not be included in the "GUI-functions".
IMO it is a really bad idea not to keep the tested functions as they are. Try to have them separated from the GUI (View) and keep exactly as they are. Invoke them from the callback-functions. Try to make the GUI "thin/dumb/humble".