The back and forth motion indicates that the problem is in the interface wiring. Below is a link to disassemble the computer display screen and expose the interface wiring cable:
1) disassemble the lcd screen: http://www.irisvista.com/tech/laptops/ToshibaM35X/Satellite-M35X-display-1.htm
Follow the instruction step up to step 5, do step 5. Now look into the right side of the trough
exposed by removing the keyboard securing strip. There will be a silvery looking cable with a white plug on the end of the wire to the left. The white connector must be pushed completely into the white socket on the motherboard. Use a pair of heavy non cutting tweezers or needlenosepliers to push the white connector into the socket. You can turn the machine on and see if it is improved. If it still is not correct proceed to step 7 and stop following the dis-assemembly.
The flat cable that runs across the LCD panel must be connected fully into it's terminator connector. If the cable will wiggle from side to side using a gentle motion then the cable must be reconnected. Examine the connector that the flat ribbon wire is inserted into. There will be one little ear protuding from each side. You must pull the ears in a rocking motion toward the cable. Use a blunt thin device, like a paper-clip with a rounded end. You may have to use two of them, one keeping one side from moving and the other to push it's ear down. When the two ears are down CAREFULLY pull on the ribbon cable. If it does not move STOP. Try moving the ears in the opposite direction, if they move continue until they are fully loose. If they do not move continue. Again try to CAREFULLY pull on the ribbon cable. Assuming that the ribbon cable is loose, re-insert the ribbon cable into the ribbon cable connector. Keep the ribbon cable square with the connector. do NOT push hard, the cable will stop at the bottom of the connector port. When the cable is in the connector you will use the blunt thin device to push the little ears of the locking bar back into lock position. Now reassemble the unit to step 5. You can turn the machine on and see if it is improved. If it still is not correct, please advise.
I hope that his helps.