You can travel with both passports to the US. See About Visas - The Basics:
My old passport has already expired. My visa to travel to the United States is still valid but in my expired passport. Do I need to apply for a new visa with my new passport?
No. If your visa is still valid you can travel to the United States with your two passports, as long as the visa is valid, not damaged, and is the appropriate type of visa required for your principal purpose of travel. (Example: tourist visa, when your principal purpose of travel is tourism). Both passports (the valid and the expired one with the visa) should be from the same country and type (Example: both Uruguayan regular passports, both official passports, etc.). When you arrive at the U.S. port-of-entry (POE, generally an airport or land border) the Customs and Border Protection Immigration Officer will check your visa in the old passport and if s/he decides to admit you into the United States they will stamp your new passport with an admission stamp along with the annotation "VIOPP" (visa in other passport). Do not try to remove the visa from your old passport and stick it into the new valid passport. If you do so, your visa will no longer be valid.
Let's look at the bold sentence, because there might be a question there. It says that both passports must be issued by the same country. Since Kenya's EAC passport is issued by Kenya, it is in fact from the same country. The EAC passport isn't issued by the EAC directly; rather, it is issued by Kenya as a member of the EAC.
There is an additional requirement that the passports be of the same type. That is, if the visa is in a diplomatic passport, the new passport used with the visa must also be a diplomatic passport. Unless you're traveling on government business, both of your passports will be "regular" passports. The fact that one is an EAC passport and the other isn't doesn't mean that the passports are of a different type.