Yes, gloves with better vibration absorption will help but there could be other things that you could do as well.
Road bikes do (generally) put more pressure on the hands, and thus wrists and forearms, than flat-barred bikes. You can alleviate this by taking conscious breaks to lift each hand and shake it out every few minutes.
Road bikes also give at least 5 different hand positions, so vary your position and grip rather than just sitting on the hoods or the tops all of the time.
You might also find that the new position means that you are gripping the bars too hard, so just try consciously relaxing your grip and seeing if that helps.
It could, of course, be that you have mild carpal tunnel issues and you'll just get numb hands whatever (as I do, after a silly day riding a motorbike from Annecy to Windsor without gloves) but the above should still help.