Slab House built in 1960 Houston TX. Drain water from my washing machine is backing up in to my bathtub. I rotor routered out the drain pipes and when I pulled out the router I found plant/tree roots on the head/spring of the router rooter. So it seems I have cracks in my drainage line. I have a pretty good idea of the location of the cracks. What are my options ? I am concerned the leaking drain pipe is going to compromise my foundation.

Where do you think the cracks are located? If it is well away from the house you may be able to dig down to the exiting main. Depending on the depth this may not be a home owner job. I have seen pipes close to the foundation exiting at as little as 18 inches. While others were 7 ft. You do not want to try to dig a 7 ft trench.

Ballengerb1 was correct. You should measure the snake where it ran into the roots and figure out just where the break in the sewer line is. Under the slab or out in the yard.
Sooner or later you're going to have to bite the bullet and replace that old sewer line with PVC.
RootX or Robics Foaming Root Killer that contain Dichlobrnic are two products you could try to control the root growth.
A less expensive way would be to call around to garden supply stores and ask for fine grain Copper Sulphate. Put a 1/2 pound in your toilet and flush it down. Repeat in 6 months. Good luck, Tom