The problem is definitely the humidity combined with how you're storing them. Lettuce and tomatoes are super sensitive to that. Normally lettuce lasts a week or even ten days if you store it right, and tomatoes last about 3-5 days depending on how ripe they are when you buy them. But on the coast with all that humidity, obviously they rot faster.
First thing is do NOT put tomatoes in the fridge right after buying them. Leave them at room temperature for a couple of days, especially if they're red and fresh. When you do put them in the fridge, put them in the least cold part (some models have a specific drawer) and don't put them in a sealed bag because it traps moisture. For lettuce, this is where it gets important: dry it really well after you wash it, and store it wrapped in absorbent paper towels inside a perforated plastic bag or a container with holes. The paper towels help a ton with absorbing extra moisture. And put it in the veggie drawer of the fridge, which is usually where it's least cold.
Another thing, check if your fridge isn't set too cold or if it has condensation problems. Sometimes those things rot everything in two days. Try to keep it around 4-5 degrees. And when you buy, pick vegetables that aren't already soft or have spots, obviously. In humid weather like yours you gotta be more picky because they're already at a disadvantage from the moment you buy them.