Answer: Dogwoods actually require an acid soil, so I would not recommend lime for the dogwoods' soil.
If they are struggling now it is probably due at least in part to transplant shock since this is the heat of summer adn they are not yet well rooted and established. The best you can do for new plants is to keep the root area slightly damp, not dry and not sopping wet. Also use several inches, about two to three inches deep, of organic mulch over the root area. Apply it in a flat layer and do not allow it to touch the trunk or bark of the trees. Do not fertilize them now, this will just add stress.
Depending on what happened to the previous dogwoods, there is a chance that there is a disease or other problem affecting the new trees. For this reason I would strongly suggest you consult with your local county extension and/or retailer to try to obtain a specific diagnosis of the splotches on the leaves.
Lawn grass is also going to suffer when seeded in mid summer. It would be better to wait until late August or early September to try to establish grass there, by then the weather will be a bit cooler. Your county extension should be able to help you with soil testing and any recommendations for lime or fertilizer based on growing both dogwoods and grass in the same area. Grass of course prefer a soil pH that is closer to neutral so you may have to compromise. Good luck with your new trees.
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