How do I fix a coarse paint job on interior doors?
Sand them LIGHTLY with 120 and then 220 grit on an random orbital sander, taking care not to burn through the paint that's already on, and then re-spray.

When spraying, lay the doors flat. Apply a THIN coat of paint. The doors can be stood up once the paint has flashed. Try to do it in a place that's out of the sun and preferably out of any blowing dust. 

It's probably going to take two coats, honestly, to get decent coverage and finish. Between coats, if you got dust in the paint during the first coast, use a "between coats finishing pad" from 3M. (You would use steel wool if you were using an oil-based product, but you're using a latex product, so you need to use a non-metallic pad.) Take your time, or you'll just waste product, as you've already seen. 

If you have to backroll to control drips, whoever is applying it is applying WAY too much product. Thin coats. THIN coats. (Although you'd probably have been mostly OK if they'd just used a foam roller instead of the 3/8 nap one.)