Question: Can you duct a standard swamp cooler into an existing homes ductwork?

Reply: Cannot use a central refrigerated air conditioning system ductwork with an evaporative sytems ductwork for a couple of reasons.
1/ The evap (swamp cooler) ducting has a larger circumference as it needs much higher airflows to give the evaporative effect which also usually serves many rooms in the home all at once.
2/ Zoning of a refrigerated system (motorized dampers) are not suitable equipment for evap systems.

Replace the ductwork. And do it properly, get the sizing of the duct determined correctly for the length of the duct runs to the rooms. Balancing of the airflow is important or the outlet closest to the evap cooler will blow off your toupee

There are various types of home heating and air conditioning systems. The type you have in your home is most likely based on the age of your home and the part of the country in which you live. Especially if you buy an older, existing home.

Gas and oil furnaces have been around a long time. A furnace draws air through duct systems in the home where it is heated in the furnace and returned back to the home through the duct system.