I have tried to fix the digitizer only. I have experience in doing the same on at least a dussin different phones. But I didn't succeed, no matter that I was extremely cautious. I ended up with a damaged LCD and small glass parts all over the place.
I bought a new full assembly with LCD+digitizer and got a better result.
Still, even if you replace both LCD and digitizer it's required that you use really good adhesive. Otherwise you will end up with a screen that lifts up at the edges.