6,000 sq feet is about 60 x 100, just to make the math easy. A bit smaller than my backyard. I sympathize with the tree = shade situation. I've got one huge old tree that I would never consider removing unless it was dead or leaning over my house. It leaves me with a very small area for growing sun loving crops.
I don't trust trees that are closer to the house than they are tall. Too much danger of the tree or large limbs falling on the house. Even if you think you can't afford to remove those types of trees you really should consider having those taken out. Dead and dying trees need to be removed also. The liability alone makes a lot of difference. Any trees that could fall or drop large limbs on a neighbors house should be removed. Those are lawsuits waiting to happen.
What kind of shade do you have? Most tree shade is dappled. You can grow a lot of stuff in dappled or part day shade. Garlic, cucumbers, leafy greens, herbs, blackberries, strawberries, raspberries, peas, and even peppers will grow and produce with as few as 6 hours of sun a day. And yes, I have grown all these in partial shade and dappled light conditions. I grew some purple podded pole beans in a spot with 6 hours of sun a day as an experiment. I got a few beans, I think the poor production was more from lack of soil nutrients and water more than sunlight. Oh, rhubarb and horseradish do fine in half day sun.
But you will have more problems with root competition than you will with sunshine. I had a raised bed under a tree in the front yard where I grew herbs and strawberries. I had a lot of problems keeping the plants from wilting in the really dry soil. The raised bed was taken out for a construction project this past summer. It was full of tree roots.