I wouldn't try to scrape off the messed up area while it's wet because you'll leave ridges at the edges of the area, which will look even worse. Leave the area to dry before trying to fix it.
To get it to dry quickly, higher temperatures and low humidity are your friends. I've used a space heater and an oscillating fan when I needed paint dried quickly: arrange the heater and the fan so that warm air gets blown past the work piece continually. Leave it running overnight: it looks like you're using a higher-gloss paint, which will form a skin quickly, but underneath will remain wet for a while.
(See here for a technique for removing drip marks.)
Airless sprayers have nozzles with apertures of different sizes. Have a few different sizes and experiment to find one that sprays at a comfortable rate with a decent pattern. Try different pressures, if the sprayer has an adjustment for that. When you're happy with what you're seeing on some test surface, then switch to the real thing.
Since you're new at spraying, perhaps forget about saving time by doing both sides. Just take your time to let the paint tack and flip the doors over. It's already a huge time saver just to be spraying.
There are ways to do it, if you insist. If you have a garage with exposed ceiling beams, you can screw in some hooks into the ceiling, and into the doors, and use string or chains to suspend the doors. I don't necessarily mean that the doors are hanging above ground, but just prevented from falling over, giving you access to both sides.
Doors often have recessed features, so you have to spray from various angles, otherwise areas in the "shadow" of the spray won't get coverage. Spray those inner carved areas first, then focus on the flat, outer surfaces.
Light is your friend! Spray in a very well lit area so you can spot problems right away, like the start of a run, or inadequate coverage. Bring in extra lamps.
Sherwin-Williams carries a door and trim enamel is quite good, by the way: the
ProClassic Interior Acrylic Latex Enamel. Is that what you're using? Even if you brush it on, it flows so well that you cannot tell. The downside is that the enamel is a little bit stinky with VOC's. Yet that is probably what allows it to dry fairly quickly, yet flow nicely at the same time.
Best Answer
The root problem is poor adhesion between coats
Something has gone wrong with the latex application that is causing it not to stick to the oil paint.
A glossy surface under a microscope looks flat as a board, like the Utah salt flats. Stuff can't stick to that, this being the whole point of glossy paint. To overpaint it, you want the new paint to be microscopically jagged, have some "tooth" for the new paint to mold into and latch on. This doesn't mean heavy sanding, I do few swishes with a Scotchbrite green pad.
Usually, the problem is homeowners skip this step. And they get away with it most of the time, because most house paints are flat. They are already jaggy.
The other likely issue is that they painted the topcoat before the oil had time to cure up and become ready for recoat. The times in the instructions must be honored, even give them a little leeway.
These two layers will never bond. There is no magic juice you can put over top of the top layer: paint simply does not work that way.
Keep trying products
Latex is a relatively weak paint when it comes to solvents. The solvent for my 2-part Urethane paint makes a pretty good latex stripper, but it doesn't attack oil paints. I have also heard alcohol works well. There are also stronger strippers.
You also need to use strippers correctly. Follow the instructions and do everything they say, in particular, cover it with plastic sheet. Lots of people skip that step. When you find the right product and get technique dialed in, it should go pretty fast and lift the latex without damaging the oil too badly. Don't try to force it with elbow grease, expect the chemical to do 90% of the work.
All this to say: if it's dreadfully slow going, change methods.