Having lived in two places that added texting-and-driving laws, I remember the traffic got a lot better very briefly after the law was enacted then it got far worse a while afterward as people began ignoring it.
I believe this is because something like this happens:
1) It's not prohibited, and people do it. Most can do it safely or know when to do it safely. Survival of the fittest has already took the low-hanging fruit off the road through single-car accidents and/or they learned not to text-and-drive due to something that happened.
2) When the law passed, everyone got really nervous and didn't touch their phones. So you now have everyone paying attention.
3) After a few months, the lowest-skilled, least-intelligent, and youngest drivers all resume doing it, but now instead of holding their phone in front of them at the top of the steering wheel where they can look ahead and at the phone in quick succession, you have them holding them down and looking away for very long periods while they're typing. Now you have them with one hand on the wheel, both eyes off the road for seconds at a time, and fumbling with the phone in a more awkward position.
So, in effect, the law made the situation worse.
The solution, as I said all along, was not another law, but to enforce distracted/reckless driving laws more widely than they have been. It doesn't matter if it's a 16-year-old texting or a 35-year-old slapping their kid in the back seat, both should be pulled over and ticketed if they're driving recklessly and distracted to the point that they're liable to cause an accident.
Some people can, under the right circumstances, text quite safely while driving. They're the same people that can operate their car stereo or talk while driving, and may even be the people that know when it is or isn't safe to do something in addition to driving.