"It is getting cold outside, so I am taking a coat."
Translation:מתקרר בחוץ, אז אני לוקחת מעיל.
Using לכן in place of אז here would sound too formal and written ?
( Or לכן has a slightly different meaning/use ? )
It would be too formal and written, that's true. Whether the meaning would be slightly different? - I feel that it would be very slightly different, but I'm not sure all native speakers would agree.