constant voltage drop at (relatively) high current
I'm sorry to say I'm completely flummoxed.

Can we start at the beginning, please?

Here's how an LM350/LT1084 works:

The way the regulator is designed is that you first decide what output voltage you want, (V1) then you arrange the voltage divider R1R2 so that with the desired voltage on U1's OUT pin the voltage on the ADJ pin (V2) is 1250 millivolts less. The current (Iadj) out of the ADJ pin will be less than 100 microamperes, so by making the current through R1R2 large enough, the drop across R2 caused by Iref can be swamped out and made negligible for most applications.

If not, then from the LM350 data sheet you can always use:

$$ \text{V1 = 1.25}  \bigg ( 1 + \frac{R2}{R1}\bigg) + \text {Iadj R2 .} $$ 

To proceed without using the formula and assuming you want 10 volts out of the regulator and that you're willing to spend 10 milliamperes through the divider in order to reduce the error introduced by Iadj to around one percent, if we want a 1.25 volt drop through R1 with 10 milliamperes through it, from Ohm's law we have:

$$ R1 = \frac{Vadj}{Iadj} = \frac{01.25V}{0.01A} =\text{125 ohms.} $$

Then, to drop the rest of the 10 volts through R2, we can write:

$$R2 = \frac{10V - 1.25V}{.01A} = \text{875 ohms}$$ 

and we'll wind up with:



So there you have the basics.

Now for your switch...

Following is an LTspice schematic which shows a single-pole 10 position switch as a series of voltage-activated switches driven by a counter with mutually exclusive outputs, ensuring that only one switch at a time will be turned on, exactly like your manually operated switch appears to function.

The counter is used to cycle through 10 states and to connect 10 different R2's to ground, changing the ratio of R1 to R2 at each step, which will change the regulator's output voltage in a planned way if the ratios are selected properly.

Notice particularly that, as opposed to your design, each switch position relies only on the ratio of the values of two resistors to determine the regulator's output voltage.