My tab stopped coming on so I opened back and unplugged battery now it works accept the screen

Welcome to Android Central! Perhaps you dislodged a connection for the screen digitizer. Open the tablet up again and check all of the connections.
 
Now the screen comes on but touch does nothing
Assume your Tab 3 is 10.1 inches. Tab 3 , 7 and 8 inches , their touchscreen cable are easier to disconnect and connect to male socket on mainboard.
Probably you did not insert fully the touchscreen cable. Make sure the line on the touchscreen cable aligned with the one on main board and it's secured with tape. Be careful with the plastic locking tab, open and close it evenly, once it's broken you have to change the main board because the touchscreen will act erratically due to loose connection
Touchscreen.jpg

Open position: lift the brown locking tab up 90 degree evenly, not the black opening.. Some people make a mistake by trying to pry open this black plastic opening of the connector and end up to break it.
open.jpg

Locking position: push down the brown locking tab evenly. Be careful as it's very easy to break if closing unevenly.
lock.jpg
 
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