I've had variations on the sticky-ness too. Overall though, I don't think washi is supposed to be very sticky - I don't think it's meant to have the adhesive power of regular office sticky tape, for instance.
I can't point you to a specific brand because there's so many around and because I get most of my washi in the form of repackaged mini rolls, but there's a couple of things I noticed while using them that might be of help:
- washi adhesive is pressure sensitive: the more you press the better it sticks. It also generally seems to cure - ie, yield a stronger bond with time. If I stick a memo into my planner using washi, I can usually remove it the day after with no tearing, but not (say) a week or two later.
- overall patterned tape marketed as washi that is made by US and european companies (as opposed to asian companies) seems to be stickier. Sometimes so sticky the roll keeps tearing because it's still a paper tape! I don't know if anyone else has noticed this pattern, but anecdotally it seems to be there
Hope you find what you're looking for!