The common mode problem can be avoided by making the experiment not differential (which coincidentally saves half the wiring) but even if you do that, you still get a large influence on the result due to the coupling between the twin-lead line and ground. It's actually a pretty good demonstration why using baluns for twin-lead is necessary; only the differential mode of the line works well, because it's not actually a two-conductor line, as it's an open line it is also coupled to ground. In my experiment (with ~6 cm wire spacing, which makes relatively little difference as the impedance of the line is proportional to the log of the spacing) I found that suspending it around a meter above ground gave almost identical differential and common mode impedances. That of course rather significantly reduces the amplitude you're going to see in the experiment.