Wires from car battery to solenoid catching fire after connecting a second solenoid
Wires overheat because of over current.  

Get out your multi-meter and measure the current in the wire. 

(BTW, you mention 16 volt spikes, yet you say you don’t have an oscilloscope. Hmmm ? ) 

If the solenoid doesn’t fully engage (pull in), then you may be drawing a continuous current of  2.94 amps. Use a wire size able to handle 6 amps (both solenoids) continuous and the wires won’t catch on fire. Then you can find  other problems.  

EDIT 1 :  You also should change diode to a diode that can handle a PEAK forward current of at least 3 amps. (per suggestion by @soosai steven). The 1N4148 is rated at 400mA recurrent peak forward current.