The sensor itself would draw next to no current whether the system is under load or not, so if it reads 8V without load it should read 8v with load, bad joint won't make a difference here.

There's no way the RX can be showing 5.5-5.3V and the input to the digiswitch is showing 4.0v so there must be more than 4v (on average) at the battery like the hand held meter suggests.

There may be some electrical garbage sitting on the line before the digiswitch under load (like a really noisy servo) which is upsetting the sensor that the hand held voltmeter is ignoring, if the two are showing different voltages when connected to the same physical points.

Can you try a different digiswitch or perhaps remove the servo's from the RX and reconnect then one at a time. Just for interest, you could try measuring the battery voltage with the voltmeter set to AC volts, ideally it should read zero but if a lot of electrical noise appears (AC voltage) it should show up something..

Cheers