I have a lot of poison ivy (or used to have - it's almost all gone now hehe) on my property of the type you're describing. If the leaves are falling off the vines on the trees that are still standing, you could simply wait and they'll be gone by next year, or you could have someone come and pull the remaining vines down.
As far as the poison ivy in the grass, that should be the easiest to remedy. Cut your grass and bag the clippings. Since you're very sensitive to it, buy a disposable painter's coverall to wear while you're bagging the clippings. Remember, you only need to do this in the area that was under and near the tree. When you're done bagging, spray the inside of the mower bag with enough Lysol to make sure it has soaked everything, and you're done. Since the grass is tall now, you should only need to do this once before the ivy oil is gone.