Do I need to get my valid US visa on my old passport stamped again?
No you need not. Provided both your old and new passports are of the same category (i.e Official/ Diplomatic/Ordinary). The immigration officer will check the validity of the visa on your old passport and put a VOOP (Visa On Other Passport) Stamp on your new one. Needless to say you must carry the old passport which has the still valid visa. In any case, always carry ALL passports that you have ever held whenever you travel. I've been asked for my old passports a few times. Happy Travels!