how can i tell if my automatic choke is staying on when it shouldn't?

it's me again, 65 bug with excessive fuel consumption, high idle after
i drive for awhile, and run-on after shutoff. i dont have an
electromagnetic cutoff jet, and my mixture isnt too lean. and i know
my choke is working, but now i wonder if it's failing to shut off. if
it's continuing to heat the automatic choke spring after i have
reached normal operating temperature, wouldnt that cause a high idle?
and the high idle cause my engine to run-on because of angular
momentum?

how could i test it? i have a voltmeter, should i be able to get a
voltage out of it even after ive been driving for awhile and the car
is hot? i dont know how it works...

plus ive been getting a dead battery after long night drives. ive
tested with my voltmeter, it's the headlights that are draining me.
any ideas there? how to check if theyre grounding somewhere onto the
car? and is there a better way to check my generator output than
testing terminal b51 on the regulator after disconnecting the battery
from it? i cant hold the meter and increase the RPM to peak all at
once, by myself.