Hi mate, try opening it up and checking the battery connection. I've had this issue before, and resolved it by making sure the connection was good. Should resolve the issue.
I would suggest checking eBay first, then Amazon. Plenty of them around.
I got mine on eBay, but there are plenty of places to buy the screen.
The proximity sensor is attached to the cable shown in step 9. You can open the phone up and check that this is plugged in properly. You only need to remove the back case & first cover.
This is all part of the ribbon cable that sits on top of the battery. Check the connections at the top and bottom of the battery.
There are actually two types of tape that I've encountered. One is easy to remove (double sided foam tape), and the other is very hard (some type of self amalgamating tape). I actually ruined the battery with the stronger tape. If I were doing it again I would recommend heating the screen before trying to remove the battery.
It might be worth opening the phone up and checking the connections - particularly the small connection shown in step 9.