when you do the close grip bench press try not to grip the bar tightly with your hands , try to just push the bar with your palms to release the stress from you wrists....doing them with dumbbells is ok but you will not be concentrating as much on your triceps because your arms would not be very stable and you will spend energy trying to keep your arms together and elbows in.
you can use the smith machine and this way you don't even have to grab the bar tight just push it with the palm of your hands: elbows in and bar going down at your upper chest.
another option is to stop doing close grip bench if it hurts and give your wrists a break for 3 weeks if they hurt, you can do something else instead
or use some wrist wraps only while you are doing the close grip bench , sometimes a supportive wrap can help ( but to be used only during benching)