Is it motherboard or screen

Hello Amanda, you did not mention what model of iPhone you have. From your description I am going to say it is likely the touch IC chip on the motherboard at fault. I've become aware of a problem with at least the iPhone 6 plus and the 6, that is known as touch IC disease. This problem is caused by the touch IC chip either failing or it may have cracked or cold solder joints under the IC chip. You being able to use it for awhile by hitting it against your hand and then after time that stopped working makes me lean towards a touch IC chip having cold/cracked solder joints. I've repaired a few iPhones lately with touch disease and noticed on one that if I twisted the back casing of the phone just right the touch screen would work until I twisted the back case again. That told me to check the touch IC on the motherboard. It was definitely the cause. Got the chip re-balled and it the touch worked great. I'm​ also thinking the touch IC because a new screen had the same issues where touch did not respond. So it could possibly be the touch IC chip on the motherboard that needs replaced or at least re-balled to get the touch screen function again. Anytime there is a hard drop the force can cause cracked solder joints on the motherboard. I wish you luck and hope I was helpful.