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It's hard for me to say. I'd recommend you to send in the phone to a service center to get it repaired. It's easier to fix a cracked screen then a water damaged phone.
Unfortunately the display can be expensive to replace, as it is pretty much half of the phone.
Hey so in my opinion its not as i have a broken back glass to and ita not waterproof, however i dont know is it the back glass or other parts not waterproof. Install a app called "WaterProof" by valentin draganescu this app is trusted because it says my phone isn't waterproof but when i test it on my tablet it says it is waterproof, and it is actually waterproof because its been in water many times, when i open the flaps it says its not. The results aren't random because the test is measured by the pressure sensor. Try that app.
Don't trust those apps only a vacuum test through your local Xperia Care could give you the correct answer, don't risk it.