You should certainly travel with both passports. It is actually very common. I work for an airline and check in passengers every day with two, even three passports containing various visas. Hardly anyone travels with their marriage certificate if it's a simple case of just having your name changed through marriage, but that doesn't mean it isn't good advice to carry it anyway just in case. Maybe get an extra copy of the certificate rather than carry your only copy.
Most people staple the back cover of their new US passport to the back cover of the old one. If you have one of the new chip passports, then don't staple but use a rubberband instead to keep the two together. Open your old passport to the page with your visa in it, slip the rubberband through there, then slip the rubberband through the middle of your new passport. This way airlines and immigration officers can easily scan your new passport and find the page with your visa in it on the old one.
Brigette