I have a furnace with model number nugk100dh08. When I turn the thermostat to heat the exhaust fan turns on and the burners light and stay on. The blower never turns on and the system shuts off after a few min. When turned on manually the blower will start up and stay running but not automatically during the heating cycle. I have tried multiple thermostats.

The furnace is going off on high limit, not good for the heat exchanger. I can't find a wiring diagram for that furnace. Can you post a pic of the diagram?

The wiring diagram is at the link below on page 84. http://icpindexing.toddsit.com/documents/086477/NUGK%20SERVICE%20MANUAL%20PART%202.pdf I think I might have found the problem. My system relay, terminal number 1 is broken and loose in the housing of the relay. I think by looking at the wiring diagram that this would cause my issue that I am seeing?

The system relay certainly sounds like a good place to start but I couldn't tell how that would effect the fan only in the heat mode. When the fan should be on, check across Brown & Yellow on the fan relay. If you get voltage that relay is open when it should be closed. If you get no voltage across Br/Y, check each side to ground or neutral.