I have been on Sprycel a year and I have the same problem. Right now, by some miracle, I am sore-free. I have been for about 5 days (the longest stetch ever). I totally empathize with how painful they are. I have deep scars on my tongue and cheeks. I ran to a dentist who then referred me to an oral pathologist. Sadly, there is not much that anyone can do. I know how horrible it is when they last for over a month. I just accepted the fact that if I wanted to stay on Sprycel (which I also felt great on otherwise and still do), that I would have to endure the pain. I saw it as, "it is better than being dead". This pain and agony had to be worth going through to see my kids grow up. But, needless to say, it totally sucks. I am not sure I ever had worse pain. I'd actually take a BMB and childbirth over the chronic sores (or should I say gaping, ulcerated and bleeding crevasses in my mouth). There are times you cannot talk, eat in public, or just eat period. I so understand.
Sadly, I found that the mouthwash mix the oral pathologist made me did not really work. It numbed all other parts of my mouth, but did little to numb the sores. Canker Cover works well to patch the sores (sold at Whole Foods Market and most drug stores). They are usually big enough to cover the large ones. You can also break them apart for the smaller ones. It helps. Blistex Kanka does the best at numbing the pain for the longest amount of time, (much better than anything a dentist has given me). However, it is very sweet and has an icky taste. It has a coating that falls off immediately, but it is worth it. There is also Zilactin B, which does an OK job at numbing the pain, but has a great sealant that lasts almost an hour (it claims 6 hours, but I am not sure on who or what). Again, Kanka has the longest lasting pain relief. Those are the best things I have found in my desperate search for relief (after the medical professionals gave up on any hope for me).
The only thing I can think of to help overall would be a lower dosage of Sprycel. If you attain a good response (i.e. MMR), that may be possible. I have also been taking L-lysine every day. I up the dosage to 1500mg when the sores get really bad (holes or multiples all over my tongue and down my throat). It seems to help a bit. I also use mouthwash to rinse and a toothpaste without sodium laurel sulfate. Healing-scents.com has a very mild toothgel. It is not as strong as regular toothpaste, but it does the job and it totally non-toxic. There is also a brand called Toothsoap.com, but I found it causes me dry mouth, which I never had before using it, but you may have a different experience. I really liked it and felt it cleaned my teeth well. You just gotta get used to the slightly soapy taste and mint or whatever flavor you choose. Neither of the two brands I mentioned are sweet. So, you have to get used to it. Oh, I have tried zinc, Maalox and the like, and DGL, but other than a good taste, (except the Maalox---ickkk), I find it helps little.
Anemia is also associated with mouth sores. So, if you are low, this could add to the problem. However, I never had mouthsores when I was on Gleevec and almost dead from lack of red blood cells (seriously folks). I believe mouth sores are a nasty Sprycel side-effect for the few of us fortunate ones.
Best of luck to you. I hope what I suggested helps. Here is to your body adjusting soon and kicking those slow moving cells in your mouth into major mitosis!!!! As it slows down leukemic cell division, it appears dasatinib slows down those of the mouth as well.