Is it worth using non-VOC or low-VOC interior paint

If you must paint this stained wood, a light sanding with 4-O steel wool or 320-grit sandpaper should be fine. All you're doing is roughing up the surface coat, usually poly, which will let the paint "key" to the surface better.

Be aware of the finish originally used on the wood; only poly will take a latex topcoat well, while on most other finishes like varnish or oil-based finishes, latex will not bond to the surface properly; it may peel easily, or even bead up when applied. If this is the case, roughing up the topcoat is not enough; you will have to sand off the topcoat to get to raw wood, and on a profiled wood piece like mouldings, this can be difficult or impossible.

Also be aware of the type of wood; certain woods are "open-grain", and will absorb paint deeply into the structure of the wood, requiring many coats to "fill" the wood and get a solid color on top. Otherwise, the grain and any knots will show through the final coat until you put 3 or 4 thick coats on. If you have to go down to bare wood on an open-grained wood, you have to seal the wood with a product that is compatible with your paint base.

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.)