I know this is an annoying question to answer all the time, but it's so hard to ever diagnose a humidity problem without knowing the answer. So, have you calibrated (tested) your hygrometer lately?
If so, a large portion of cigar smokers have found they prefer their cigars at a lower humidity (typicaly around 65%). While there is nothing necessarily wrong with 70%, many havefound that they run into the problems you are mentioning at that level of humidity. This is particularly the case in the Cuban Cigar community.
This probably doesn't help a whole lot so here's a couple of recommendations:
1: You can try switching to 65% (or something lower than 70%) and see if it solves your problem.
2: And this is not a bad idea no matter what. Clip the cigar you are planning on smoking and test the draw before lighting. If it's tight, put it back. Come back to it another time. Most times this will fix the problem. If you go back a couple of times and it still doesn't help, try...
3. Dry boxing. Get an empty cigar box and throw your nex to be smoked cigar/cigars in their the day before. If it's the humidity that's causing your problem, but you don't feel comfortable lowering the humidity for storage, this may solve the problem for you.
Hope that helps.