There is no right or wrong in this, but if you don't feel an inprovement in pain levels within 2-3 weeks after botox, it's likely because it has failed.
My burning was pretty much gone right after botox. It was replaced with soreness, fullness, shooting pains, irritation, itching and what not, but the horrible burning was gone. I was happy with the "trade".
From reading others experiences, I reckon this might have been a bit unusual, so don't loose faith just because you still have some burning pain. Going from 10/10 to 3/10 is a great improvement, and should give you confidence in botox is working its magic.
I'm sure you will see even more improvements if you can get consistent soft BMs, so keep up the good work!