5500 Watt / 30 amp generator feeding 5000 Watt 20 amp manual transfer switch
From the information you have given out on your last comment "maximum 5000W" the answer would be that you need to replace the transfer switch to match the new generator. 

Your old transfer switch (xfrs) has a max rating of 5000W and you new generator actually has a surge wattage of 6875W which is considerably more than the existing xfrs rating. We are not taking into consideration what is connected to your current system. It's a matter of how much the generator can provide. In short you size generators to provider a certain amount of power. You use what's connected to calculate the required size not the other way around.

Good luck