Battery needs renewed due to age.
Starter button needs replaced as you are getting 12volts on both sides without engaging button. This is causing the starter to run "by itself"
If you want to check the cables for voltage drop "put the meter leads between the battery post and where the cables connect either to the starter or the ground point." Then engage the starter and the meter will read the voltage that is dropped between those 2 points. I would look for .5 volts or less to be dropped.

 

I picked up a new switch yesterday and a new battery today or tomorrow

 

There should be a starter relay between the switch (push-button) and the starter motor solenoid. This small relay can overheat and respond with a delay. In such case it will not stop sending 12V to the starter solenoid as expected.

If you disconnect the starter relay once the engine runs you isolate the starter solenoid coil. The output_voltage on the (lower) starter solenoid contact must return to ground as soon as the coil is no longer activated.

A rule of thumb:
- Your little switch can handle 0.5 Amp to power the relay,
- the relay (inside the fuse holder) is designed for 25 Amps to power the solenoid,
- the solenoid will easily handle 350 Amps or the full Cold Cranking Amps of the battery.

The starter motor is 1.6 KW or 160 Amps under 10 Volts. A cold start will briefly draw 2.5 times the nominal intensity >> 160x2.5 == 400 Amps