It seems to me that drunken is often put before a noun, eg, drunken driving, a drunken sailor, a drunken man. In contrast to it, drunk is often preceded by the verb be.
When I said '[becoming] less common' I didn't mean 'dying out'. But I remember a time when - for example - the BBC news didn't refer to 'drunk driving'. Now it's either 'drunk driving' or 'drink/driving', and people who - in the UK, at least - still say 'drunken driving' tend to be 'of a certain age' (like me).
"Drunken" as a metaphor for "not very well controlled" is still widely used: 'even at this time of year, a few drunken wasps are still flying about'