For the last few months with the weather being so cold our furnace has run almost constantly and for the last month it has been acting funny. It will kick on for 2-3 minutes and then shut off without the fan ever kicking on. We think the condensate trap is clogged but not sure how to flush it since we can't find a manual for a 1984 model Carrier furnace. Any help is appreciated.

After 30 years, be glad that the drain is the only issue. Can you get to the drain? I've seen HVAC tech take a hose and blow out the drain from inside the unit. The trap should be removable or should have a plug or cap to remove to push a wire through.
That being said... the condensate drain is for air-conditioning, not heat. When condensate fills the pan, it will shut off the unit. (on some models). I believe your problem lies elsewhere... like the fact that your furnace has already been working twice beyond it's life expectancy. Old coal furnaces and boilers can last 50 years... but newer units living much past 15 years is pretty good... especially gas fired units.