With respect I have not been to Zakynthos, but at the same time do recommend some Greek Islands. I know better Crete. That said.
Fully inclusive for the two of you, you will get a reasonable deal, as Fully Inclusive is based on at least 2 sharing a room.
There are some Greek Islands this year, that are unpopular because of the amount of immigrants that are reaching them & Greeks seem to have no problem with these people, but some of the tourists are uncomfortable with them, BUT it is limited which Greek islands the mass of the immigrants are reaching.
in the main if you are in a Fully Inclusive complex within Greece the immigrants will not affect you. What I can only suggest is if you decide on Greece, then search you which complex and then ask if anyone has been & can recommend for you two.
What you need to take on board, is what your requirements of Fully Inclusive are?
Often the hotel complex may have a number of swimming pools & although not on the beach often, but not always has a free bus system to a beach. I would also suggest if you are choosing Fully Inclusive, perhaps better to choose a complex that caters more to fully inclusive or totally to fully inclusive, rather than a complex that it is one of the choices.
Look forward to hearing more of your choice or requirements. Greece island, you will get the sun and relaxation.