Hi Christina, good that you are aware of this at least because tree roots are barely visible, yet remove such a lot of moisture, especially ash trees!
I would cut the trees: here for example, ash are really giant weeds: but it is not my plot! I cut two field maple in 2009, which were making a big difference to nearby beds in summer, and the hedge has already replenished with hazel from old stools.
Large trees can quickly make new roots if you chop them. Even digging is only a temporary respite, until midsummer or so. You could grow winter salad…