Oak trees can have issues with TOO MUCH shade and too much leaf drop. How far up have you trimmed out your trees? Here in the panhandle we get hot dry summers and we are so far over that noon is about 2 pm by the clock during the summer, so it starts to heat up later in our day. For tomatoes I have found that afternoon/evening shade helps with keeping them happy in a hot summer--a lot of the heirlooms don't like much above 85f or cooking in the afternoon sun. Lettuce, put them in the shadiest part you can find, and try growing some heat tolerant types. Suggested types-Bibb=Buttercrunch, Speckles, Summer Bibb, Summer Bibb Blend (I haven't tried any of these) Crisphead=Michelle (haven't tried this) Leaf=Black Seeded Simpson (fair, still can bolt on you fast), Green Salad Bowl, Red
Deer Tongue (the other two haven't done well here) Romaine=Little Gem, Parris Island Cos(I swear by this one. You can pick it as a leaf lettuce and it does reasonable. Replant often).
Map your shade and towards the edges of the shade pool you can plant the more sun loving ones and seek the shadiest parts for things like your lettuce. Good luck.
