That happened to me when I began oil pulling 15 months ago. I began with sesame and added some oil of oregano in a couple of weeks later and got an enormous sore on the bottom of my tongue...it was about an inch long and .5 inch wide -- very painful and lasted more than a week! I attributed it to the oil of oregano, although it may well have just been the OP process.
I went to my doctor at the time (for an unrelated reason) and showed it to him and he said that canker sores, in his opinion, were just cold sores (i.e. herpes simplex) inside the mouth. (I have never had cold sores although I also read that HSV is generally dormant in most people.) I did some scouting around on cold sores and oil of oregano ("ooo") in particular, and found a couple of old posts here that said that ooo would push a dormant virus up and out of one's system. I'm not sure if that is true but I have not had a recurrence of the sore since then (thank goodness, because it was incredibly painful and in fact kind of insanity-inducing!).
Since then, there have been several posters who have reported mouth sores after beginning OP. One idea last spring was that it was a very strong detox reaction -- when the body can't expel toxins (either viral/bacterial, or stored metals, etc.) normally (i.e. via the kidneys and liver), it will "send" them out through the skin -- in this case, the very sensitive and painful skin of the mouth -- just thinking about that time makes me shudder!. I remember someone suggesting that it might have been fluoride detox, for instance.
In any case, a frustrating thing about OP is that people react so differently to it and you and I seem to be one of a small number with this reaction, for instance. I remember being very confused about why exactly this awful thing had happened and what to do about it/expect in the future. One thing to do is to make sure the liver and kidneys are functioning well with some liver flushing/salt-water loading. Another thing to do is to stay on a very conservative path with OP for a while -- maybe 5 minutes at a time with a break between days, as is your plan. (My feeling has always been "better out than in" as many say around here -- if something is inside my body and producing such a reaction, I would eventually like it to be gone!) I was probably a victim of my own early enthusiasm for OP and was doing it quite often (and adding the powerful ooo) -- I wanted to be "clean" fast, and didn't really take into account that over the years I have ingested many toxins that might take a while to leave the body. This is why I pretty assiduously recommend to those new to the OP process that they follow the basic protocol set forth at http://www.oilpulling.com
(sesame or sunflower, as you are doing, 1x/day for 20 minutes (or in your case, as you said, 1/x every three days for perhaps a shorter amount of time). (I have also read about how powerful sunflower and sesame are as detoxers, so one thing you could try is changing to olive oil or another oil for a little while to see if that makes a difference.)
I think part of the reason I was attracted to OP was that it seemed very easy and harmless overall, and I have come to believe that it's a great therapy on several levels (not only physical but mental as well, for instance). I am glad I stuck with it after the 10 days or so of my mouth sore nightmare. BUT, I have also developed a deeper understanding of the power of this deceptively simple therapy by seeing both the tremendous benefits people have reported and that I have experienced, as well as by experiencing and reading about some of the early detox stuff like you are experiencing now.
Finally, to treat the canker sore I was told to use cayenne pepper directly applied, which seemed insane to me at the time, but which I would do now in a heartbeat after more experience with cayenne and its painkilling properties. I used a combination of lidocaine and benadryl as a rinse, which did not stay on/do much. I also began adding lysine, which is preventative against canker sores. (I learned, too, if you are using toothpaste with SLS, that it can cause sores.) I finally resorted to narcotic painkillers which, while they probably didn't help the sore go away any faster -- Tylenol being such a liver-clogger: I always break out when I use it! -- certainly made me feel a lot better about my situation (whereas when I got the sore I seriously contemplated going to the hospital and begging them to admit me and knock me out until it was gone, and also wondered if I knew anyone who could procure some cocaine for me -- the ultimate numbing agent. I mean, I was *miserable*!). And there is a product called Kanka which is a brush-on numbing agent (also lidocaine, I believe...) which was semi-helpful.
My heart goes out to you, but I assure you this will pass! And hope you can figure out a way to comfortably OP by building up over time. Please write back if you have any questions about what I have written, I am sure it's disjointed since I have been running up and down the stairs to write/get my kids dressed, etc.