but we don't even have do that and infact we can run a differential equation(ODE) directly in software with a little thought.

Treating a digital filter as a numeric simulation of an analog filter is certainly possible, but I think I've mentioned why this might not be the best approach and why more "advanced" ODE solvers (basically, anything other than the explicit Euler method) are completely unsuited for signal processing applications (mainly: You do not have the input values for fractional samples).

 In_filter:
Dti = Adc_present - Adc_last ;test charging/discharging
If Dti < 0 Then              ; if neg. then exec. following code
Dti = Adc_last - Adc_present
Dti = Dti * 0.35             ; calibrate filter to secs
Dti = Dti / Fil_time         ; fil_time=cr time (adjustable)
Adc_last = Adc_last - Dti    ; form final output
Gosub Disp                   ; display it
Return                       
End If
Dti = Dti * 0.35             ; if discharging then go here
Dti = Dti / Fil_time
Adc_last = Adc_last + Dti
Gosub Disp
Return

Still, why do you distinguish between charging and discharging ?
The following does the same job:

 In_filter:
Dti = Adc_present - Adc_last ;test charging/discharging
Dti = Dti * 0.35             ; calibrate filter to secs
Dti = Dti / Fil_time         ; fil_time=cr time (adjustable)
Adc_last = Adc_last + Dti    ; form final output
Gosub Disp                   ; display it
Return 

How would I adjust it as outlined in 1 and 2 using your difference equation?

Well, let's look at your equation:

Adc_out_new = Adc_out_old + (t_sample / fil_time) * (Adc_in_new - Adc_out_old)

It can be rearranged to:

Adc_out_new = ( 1 - (t_sample / fil_time) ) * Adc_out_old + (t_sample / fil_time) * Adc_in_new

... from which it is quite easy to see that a in my equation corresponds to (t_sample / fil_time) in yours.