Your problem may not be the carb!
The 360 has a rubber throat/'boot' between the carb and the cylinder. Your saw is old enough that the rubber may have a tear/hole in it causing a 'carb' like problem.
To see the boot, remove the carb, release the rubber throat and split the saw by removing the 4 A/V bolts, the bottom handle and the oiler line. Be sure to pull the A/V plug behind the bar to see if the internal webbing has been torn.
Bill Walcott
785-562-3539
Servicing HOMELITE for over 50 years...
Bill Walcott @gmail.com, December 2007