Finale is doing what it should do - transposing the Eb part up a sixth and adding three sharps. I assume you're entering the score in concert pitch? What you could do is go to the Transposition drop down in Score Manager and set the parts to No Transposition.

I haven't been writing it in concert pitch. This score is a reduction from a larger ensemble so I have to bear mind to the actual instrument ranges as I write, so I'm writing in each instrument's written pitch. 

However....I believe that may have solved the issue. Should work for my purposes anyway!

What you probably need is the "Keyless" option which exists in Finale v25 but I do not think was included in F 2014 versions. 

In version 25 it is a pulldown beside the home letter in the Key Signature Tool where you can choose Major, Minor, Keyless, or Nonstandard. You might check to see if Finale 2014 has it.

I was using the "Keyless" option, but still stuck with this situation. Not sure if it's a bug or what, but no matter what I've tried other than the above solution, I kept getting stuck with accidentals if I didn't have a key signature

I stand corrected...after making that change it's now writing in concert pitch, so I'm back to where I started. Am I missing something with the keyless feature? I don't understand why it isn't possible write notes without accidentals unless you ask for them