I started single side feeding about two weeks ago (my daughter is 7 weeks today). I would highly recommend it. I actually found out about it when I did an internet search for "colic and breastfeeding" to see if something in my diet was causing her to be fussy (it was just that I wasn't noticing her sleepy signals). On of the sights said to try single side feeding if they were eating too often because tehy would get the higher calorie milk. It also said that your body would get used to it, so you wouldn't have the engorgement issues.
Instead of how Tracy suggested to switch(I just got this book), I had her eat until one side was empty, then had her finish on the other side. The first couple times, I was somewhat engorged, but this quickly faded.
My favorite reason for single side feading is that I no longer feel like I am going to run out. Before, I had several times that I gave her formula when she was still hungry because I was OUT. Now, I don't have to worry about if I will run out or not, because if she demands more than a usual feeding, I know I will have extra on the other side
Another great thing about if my right side is full, but I am feeding on my left side, I can pump my right side, and have a full serving!
After knowing the facts of the breastmilk becoming higher in fat the longer they eat, it only makes sense to me to breastfeed. If you feed for ten minutes on one side, you barely get the second form of milk, let alone the third.
As far as you worrying if you will run out, I will say you have less chance of running out feeding on one side each time mostly because you have the other side to resort to.
Good luck with your choice!