A couple of things come to my mind. From my experience and if I understand it correctly, if you antenna has a good RF ground to work against, then you will see very little if any, common mode currents on your coax during transmit. If you are seeing RF on the shield of your coax, either the antenna has insufficient RF ground and using your coax shield and anything else its connected to (not good) for RF ground, or the antenna is designed to use the coax braid as an RF ground as in a NGP antenna. In my honest opinion having built all of my own antennas, the first thing you need to work out is your RF ground, whether it is radials, counterpoise wires, a ground rod, metal roof, gutters, etc.. Also if you measure a 1:1 SWR at the antenna, and the antenna is working right, you will also measure a 1:1 with an analyzer with the coax hooked up. Remember, ANY time moving, touching, changing length of coax, etc. changes your SWR, your coax is part of the antenna. Solve this first and your antenna will get more signal in the air for you.
I just though of something else, are you running a tuner?
...runna pile-up on 6m SSB
W4AXW in the air