AnalogRead from voltage divider ladder
The datasheet of the ATmega328P (the main chip of the Uno) states:


  The ADC is optimized for analog signals with an output impedance of
  approximately 10&nbsp;kΩ or less.


The Thevenin equivalent output impedance of most taps of
your voltage ladder is higher than 10&nbsp;kΩ. You 1&nbsp;MΩ pulldown is
way beyond that limit. Cross talk from previously read channels is
exactly the symptom you expect from excess output impedance.

Taking each reading multiple times, until the value stabilizes, should
help. A buffer (e.g. an OP-amp set as a unit gain amplifier) is the
usual solution to this kind of impedance problems.




  Actually, the whole works perfectly with R as 220 and pull downs as
  10K. The problem comes when connecting more than 3 strings at the same
  time, on any tap, as the pull down resistors become part of the
  ladder (i belive).


The pull down resistors will indeed have an effect on the potentials of
the voltage ladder, especially if you connect several strings. This
could be minimized by having the pull-downs be a lot more resistive
than the resistors along the ladder, but then you would need very low
resistances along the ladder, which is not desirable because of current
consumption and heating.

Here is an idea for an alternative solution: If you can spare 6 digital
pins, you could build for each analog input a pull-down circuit that is
switchable between a “strong pull” mode and a “weak pull” mode. Assuming
D0 is the “pull strength control” pin for A0, it would look like this:

A0 ---+--- 1 kΩ --- D0
      |
     1 MΩ
      |
     GND


Whenever you want to read A0, you:


set D0 to OUTPUT LOW, which creates a strong pull-down to ground
take a couple of analogRead()s
set D0 as INPUT, which leaves only the weak 1&nbsp;MΩ pull down
take a couple of analogRead()s, keep the last one.


If A0 is not connected to the ladder, then the first reads will
discharge the internal sampling capacitor of the ADC. The final read is
done with only a weak pull-down, but this should be OK since the
capacitor has already been discharged.

The potentials along the voltage ladder will be disturbed by any string
connecting a tap to its strong pull-down, but this disturbance will
disappear when you switch to weak pull-down, so it will not affect the
readings.