However, whenever I walk into the kitchen in the direction of his food bowls he follows me and continually begs for food. I have tried giving him snacks to appease him and hold him over until meal time, but it doesn't stop his continual crying for food. My poor Sam seems to always be hungry, yet he's so overweight. Dear Human, I would try free choice, low fat kitty food for Sam. Sam will still cry for food - you must be tough.
Overweight kitties can get sick, just like overweight humans.
I am not making this up - that limiting the food of a fat cat can kill it. So let Sam eat all he wants of the low fat dry food. Also, see if you can increase his exercise level by playing with him more. After Sam is used to free choice dry food, if he is still too fat, you vet might have some additional ideas for reducing his tonnage.