I've developed fret buzz on the fat E string on my Dean Tradition acoustic. I can fret the string at the first fret and the buzz is gone. Is that an indication that the nut needs replaced or repaired?

I tried to eliminate the buzz by screwing around with the truss rod, no joy. What do you guys recommend? Bear in mind this is a cheap guitar ($100 for guitar and case).
What guitar? Truss adjustments are for neck relief to prevent buzzing in around the 4th to 10th frets behind the string. Better not to mess around with things until you identify the problem. The first thing to consider is the nut since fretting the string eliminates the buzz. Maybe the string gauge is too light (small) for the slot. Maybe the string does not angle down enough from the nut to the tuner. Maybe the slot is cut too low. Maybe the first fret is not level. Maybe that string is low at the bridge. You could get an opinion from your local guitar store.