I feel your pain. I have the same problem and the trees are my neighbors so there isn't anything I can do about it.

I put soaker hoses in the beds that are most effected. It helps some but really the poor plants dry out really badly and are stunted because of it.

Can you dig the dirt out and lay down a barrier? There are root barriers, stronger than weed barrier.

My beds are too big for this.

It may be the shade from the trees or another problem that is "killing everything". Theoretically you CAN deal with the roots by starting over with a root barrier under the vegetable(?) beds, if there is a product available that deflects roots but allows water to drain through. If you lay down something that isn't permeable of course there will then be the problem for both your plants and the trees of drainage having been impeded - this is not likely to work out unless you would be growing bog plants that could take the shade from the trees - and the trees were not bothered by being cut off from rain and air in those sections.

Some vegetables can be grown in some shade, see Schenk, THE COMPLETE SHADE GARDENER (Houghton Mifflin) for discussion. But in general you want to have vegetable beds located in a bright and sheltered (non-windy) location for optimum results.

Shade is not the issue. The trees are on the north and west sides of the garden so while they do affect the amount of rain the garden gets, they do not shade it. I may need to try and find some of the root barrier.

A root barrier is installed vertically. Must be about 3 feet wide -- look for the kind called "bamboo barrier."