Most phones today have a screen that actually requires you to take the entire phone apart before you can replace the screen, so I would certainly caution you against trying to fix this on your own since you will most likely need some tools most people don't have laying around their home, like a soldering iron.
You can get the screen replaced for a fee, computer shops in my area service cracked screens. However, it sounds like a sensor is malfunctioning and I'm sure that the screen alone will be the only fix. Like mentioned above, you will most certainly need to completely disassemble your phone to take the screen out and the crack may actually be pressing on this sensor causing the phone to think it's constantly on your face... Not certain as I don't have your phone, but I would definitely call around to see if it's fixable.
If it's not able to be fixed locally, just buy a phone online that is open for your service provider...this phone will most likely be stolen, unfortunately and runs the risk of being shut off if discovered it was stolen.
To summarize, don't break your phone.
