Today, I noticed that the water in my basement floor drain in the laundry room was higher than normal. When the toilets were flushed, washer drained, or shower taken, the water level came almost up to the top of the pipe. I augered the drain out with an electric auger with the spade tip and the spring type one. I got roots, a clay like substance and even something that looked like a piece of small material. I went the full 75 ft of the machine.

The house was built in the early 1950's and also has an outdoor drain in the basement stair entry way. The outdoor drain also is going up and down slowly like the indoor drain when water is being drained. Even after rodding the drain out, it drains so slow that now when I use the washer, the drain water comes up and floods over the floor. I had to stop and start the washer multiple times, otherwise I would have had water all over the place.

What else could I try besides calling a plumber?

How far to your septic or city sewer. 75 ' sounds like a lot but I'm 125' to the city riser. In the 50s PVC was invented yet so your drain is likely cast iron and they can fail for several reasons besides settling. Tree and brush roots can get into the tiniest crack and then grow and swell like there's no tomorrow. Rod all the way to the city riser and if that doesn't work call a plumber who has a SeeSnake, a rodder with a camera, you get to keep the VHS he makes of your pipe.

Try using the 2" cutter blade on your snake, if possible, OR a single 3" cutter blade (see pic.)

If you used an arrow or boring gimlet they won't necessarily grab onto the roots for you to pull back.

You are not trying to cut the roots as much as you are trying to grab them and pull them back. The cutter blades just do a better job at grabbing and wrapping the roots around itself... without pulling off... which the retriever head is famous for. Anyway, just wanted you to see pictures of other blades for the electric snake.

If possible, see if can get a snake head like shown and try again... otherwise, like BALLENGER said, time to call in the pros, have them snake it and then camera it... they can best advise after that. Good luck.

If it's a root problem I may be able to give you some relief.
RootX or Robics Foaming Root Killer that contain Dichlobrnic are two products you could try.
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.
Hope this helps Tom