Sure:
View attachment 118543
I added a resistor, zener diode and capacitor (on the left) to hopefully get rid of voltage spikes from the alternator. The transistors that are driving the LEDs (terminals marked with "G"/"D"/"S" - i.e. gate/drain/source) are N-channel mosfets; choose some with voltage rating above 20V, and current rating above whatever the LED strips use.
The 4093 only has one pinout that I know of, which is the one you posted first.
It's just as simple using two 555s. I can draw the schematic if you want.