Your best bet would be to stick with a 36" tire. Once you go over that it is hard on the TTB, but for an on road truck it should live. I put on a six procomp lift with 38" tires (sawzall helped the clearence issues) I had no problems on road, did wonder more than I liked, but first off road adventure snapped driver side axle. And later that day broke passanger side axle. I really don't see much difference between 36 and 37, except price since a 37 tire isn't as common.
With 37 you'll have to adjust your bump stop. I rub pretty bad when I turn. And you will have some trimming to do. Unless there is some reason for running 37", I honestly recommend to just stay with 36" tires. It is easier and their is a wide range of choices.