I decided on Saturday night that I needed to get a "power auger" or "roto-rooter" to do the job. I found out that the Home Depot rents them. I went to HD and an associate named Big D helped me get the equipment. All the 75' rooters were rented out, so I took a 50 footer home. I felt it take care of one clog, but that apparently didn't do the job. I couldn't back feed it from the clean-out in front of the house, so I had to go back to HD and get a 75' machine which had been returned in the mean time. I bore through the remnants of the first clog and powered through a second. That fixed me up. It was a $45 fix which is not too bad considering the alternative of hiring a real plumber. And now I have that life experience that I was lacking just a few short hours ago.
Let there be no question that roto-rootering pipes is not the most pleasant thing in the world. You get that sewer smell in your nostrils and maybe even on your body if you're not lucky. However, there are plenty worse things (like having water back up in your pipes and render your clothes washer inoperable). My mom asked me if it would have been easier just to call a plumber. Of course that would have been easier, but it would also have been much more expensive ($75 minimum and probably upwards of $100). Overall, I'm happy that I ended up being able to fix it myself.
My laundry had been piling up before the giant clog