Worked on evaporative (swamp) coolers for 30 years, and the most notorious problem I found was not enough air flow.
I have a gas furnace with ducting (which the swamp cooler also blows into), which works fine.
The heater duct will not give enough air flow for a swamp cooler but will work for refrigerateda air. Duct work for an refrigerated air unit will not give enough air flow for a swamp cooler.
Your choices for a remedy will be a window swamp cooler, a swamp cooler that just dumps into one area, or install seperate ducting for the swamp cooler.
You hear people say, "crack your windows." Cracking your winows will not give near enough air flow.
For proper cooling with a "swamp" cooler you need to change the air in your house, or room, every 1 1/2 to 2 minutes. With a ducted swamp cooler you can open windows in different rooms to control the flow for those rooms, if the door is closed, or install upducts in each room.
A house of 1200 sf with 8 ft ceiling has 9600 cubic feet in it. To exchange the air in 9600 cubic feet space every 2 minutes you would need 4800 CFM cooler. That is a lot of air to let out of your house, and cracked windows just won't do it. Exhaust air should be around about 500 CFM per square foot. An exhaust of air for 4800 CFM would be around 9 sf total exhaust area. The air will move to where there is an exhaust hole.
I use a 4800 CFM cooler, ducted to all rooms, on my house and I normally open a window in the dining area, right next to the kitchen, that is 34" X 36." Works great except when the humidity is up around 25% or more.