Based on the info you have given here, I am going to make some assumptions. In "cleaning out all the components" you cleaned the air filter or changed it, cleaned the duct-work, and the intake combustion chamber? When the "furnace stays lite" the pilot and the main burner stays on? Lastly, when it "continues to blow cold air" is this when the main burner is on after long periods of time? If the answers are all yes, then maybe some of the ducts have come off, or the burners aren't working correctly.
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