This recently happened to me, I guessed that the left half digitizer ribbon cable became dislodged because the right half of the screen still worked. I watched a 6 minute video on how to get to the replacement digitizer screen, and you have to dis-assemble everything to get to the screen, it is the last piece in the project. I recommend you buy a new D2G, they are out there, it's a 956, do not buy a 955, that's a Droid 2. Or go to your Verizon store, most of the techs know a good cellular phone repair guy to recommend. I estimated that buying a new screen and labor charges would be around 140, and I bought a new one with old software for 168 with free shipping. It came with the old .607 sofware, so I can root with Pete's Tools, and disable updates and tether and keep my grand-fathered 3G data plan until I break or drop my phone again. If you can upgrade, do it, but if you go from 3G to 4G LTE, whatever that means, you will lose your "unlimited" data plan if it is "Grandfathered" sometime in July - August.
My Droid2 had the same problem... the phone started to freak out... apps opening and closing on its own, etc... It's due to moisture. Here's how I fixed it which will cost you about $5 and time to drive to a florist.
If you have florist or an arts & crafts store nearby, get some desccant and place the phone in an airtight container packed with the desiccant. It'll absorb all the moisture from your phone. Desiccant is silica gel pellets used to dry flowers. For maximum efficiency, heat the pellets in an oven (not microwave) to remove any existing moisture from the pellets. Leave it in the container for about 48 hours to thoroughly dry out the phone.
You can also use dried rice but you'll have to leave it in the container for about 5 days instead for thorough moisture absorption.
Hope that works!