This is an indication, that your antenna is bad matched with coax cable.
As result your coax cable works a part of antenna and this is why it's length affects VSWR.
You're needs proper match between coax cable and antenna with balun transformer to isolate your cable from antenna and use RF chokes on the cable to reject induced interferences on the cable. With proper matching and RF chokes it helps you to significantly reduce noise interference from nearby electronic equipment and VSWR will not depends on cable length.
No, that is not correct -- it is perfectly normal for mismatch to occur within a cable, and have no radiating or CM consequence. As profdc9 noted, outer shield current is only due to imbalance: asymmetry in the antenna itself, proximity to nearby structures, a radiation pattern that includes (or gets reflected back onto) the feedline, etc.
Note that mismatch should be evident on the VNA: impedance will lie somewhere on a circle centered on the origin, the angle varying with feedline length. This is on top of all the circles (across broad frequency ranges) due to the antenna's resonance and whatever, so keep in mind how this needs to be measured: plot a point at resonance, change the cable, plot another point at resonance, etc.
Further counter-example: consider driving a 100Ω twisted pair with one terminal grounded. The common mode voltage is Vin/2 by definition. This launches an outer-shield or CM current into whatever the surrounding impedance is (I = (Vin/2) / Z)), where "outer shield" in this case refers directly to the balance of currents in the wires, but applies equally well in the case of coax (where the one-terminal-grounded configuration just happens to correspond to the low-CM case). The far end of that twisted pair can be perfectly terminated with a 100Ω resistor, SWR = 1 anywhere on the line, and still the line acts as an antenna, because you've coupled 6dB into the common mode at the feedpoint. You might (rightfully) complain that this is a contrived and erroneous situation, because obviously you use a balun on a twisted pair TL -- but that's just the thing, your criteria didn't require a balun, and a properly balanced antenna feedpoint won't do that anyway.
So, just to clear that up, as no one else seems to have dug into this particular point so far.
As for mitigation, note that series chokes can only reduce, never eliminate it. Much higher attenuation can be had by pairing the series choke with some shunting impedance: tie feedline shield to ground every once in a while, to make an equivalent ground circuit of a ladder divider, with the series spans being either blank line (still has some CM impedance) or chokes (whether coiled up or ferrite loaded, the ferrite of course introduces loss that dampens the otherwise resulting resonant modes), and the shunt elements being ground clips (cut off the outer jacket, add a spring clamp), inline bulkhead connectors (gets a ground connection, maintains environmental protection), etc., to whatever nearby large metal structure (poles, beams?), tower, (metal) wall, or stake into earth, is available. What counts as "ground" should be as wide as possible: a tower is better than a pole; a grounding grid attached to a tower base is better than single stakes into the ground; etc.
Grounding the feedline has the same effect as ground stitching vias on a two-layer PCB, keeping a good ground around a microstrip trace -- or even more particular, suppose it was not just a microstrip trace but one with a signal conductor on top.
You can never eliminate the support/feedline/tower from the antenna, of course, whether in terms of reflection off the nearby ground plane (earth itself, or a large structure the antenna's installed on), or that the width of the line/pole/tower has some (even if slight) effect on radiation pattern, balance, etc. But these general guidelines can make it good enough not to matter, and for the rest, standardizing on a support dimension (width and height, and assuming a good ground beyond it) at least fixes the remaining variables.
Tim