You really need nitrogen rather than air assist to cut cardboard to minimise charring
However if you don't want to use an inert gas, try this : set ppi low , just above discrete perforation and use strong air assist - mask the material with paper based low tack sign vinyl application tape to stop surface marring. The freq/ppi is how many holes the laser drills in a set distance - the laser cuts by drilling lots of holes , closely spaced so they overlap , too little makes cuts like perforations , too much overlap puts a lot of heat into the material leading to charring.
You might of course just have too little power to cut this stuff cleanly , lasers work by vaporising and not "burning" , too little power density leads to burning. Think of making a hole thru paper with a butane soldering torch or a bic lighter - both will work but the hotter and thinner torch flame will make a clean hole and the bic's hole will be messy.
Rodney Gold, Toker Bros trophies, Cape Town , South Africa :
Roland 2300 rotary . 3 x ISEL's ..1m x 500mm CnC .
Tekcel 1200x2400 router , 900 x 600 60w Shenui laser , 1200 x 800 80w Reci tube Shenhui Laser
6 x longtai lasers 400x600 60w , 1 x longtai 20w fiber
2x Gravo manual engravers , Roland 540 large format printer/cutter. CLTT setup
1600mm hot and cold laminator , 3x Dopag resin dispensers , sandblasting setup, acid etcher