It should be blowing cold as long as the compressor is on. Nothing to do with the door open or not
Keep in mind it is possible to turn the fan on without the compressor or to have the fan blow all the time.
To test. Set the thermostats as low on it will go. It should be blowing cold. Noticeably cold.
Go into a different unit and turn the ac on to compare.
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2009 F350 CC LB 4WD
2017 Jayco 355MBQS