After all of the rocks were removed, I sprayed them off with the power washer. I wanted to get and weeds, grass, and dirt off of them before I laid the new barrier down. Since I am hoping to not have to do this again I bought the professional-grade weed barrier that was good for 20 years. I bought only one 150 square feet roll and it was just barely enough.
Note: I did not remove the old barrier, I sprayed weed killer on the remaining weeds and laid the new barrier on top.
I planted a few new plants before moving onto the last step. Which was putting all of those rocks back! Thankfully this time my aunt and mom came over to help me. We were able to get them back in place in about an hour.
Now the courtyard flower beds are cleaned and beautiful. Phase two complete and onto the third!