What’s the best way to touch up spray paint

If the present paint on the doors is indeed sticking very well and the surface finish is conducive to overcoating then I would recommend a process to fill in the peeled out areas with a spackle or drywall mud. Apply one of these materials with a putty knife, let it dry and then lightly sand to get a smooth even surface. Sometimes it is necessary to apply a second coat of the filler.

You can perform some tests of how well the existing paint is adhering to the wood. Near the existing chipped out areas look to see if the nearby edges of paint would easily pop off if picked at with a knife point or other sharp object. You can also use an aggressive tape such as Gorilla Tape and see if a quick removal of the tape would peel away additional paint. If peeling is apparent then you will have to remove the old paint.

If it turns out that you would have to remove the paint completely from the old doors you would likely want to consider the use of paint stripper. This is a jelly like liquid that you apply to the paint with a brush. It will soften the paint so that it can be easily scraped off. It may require several applications of the paint stripper to get all the paint off - especially if there had been multiple coats previously applied. Steel wool can be used to remove softened paint on edges, corners and curved areas where a scraper does not work very well.

Paint stripper is pretty noxious stuff so use in well ventilated area and wear applicable protective gear to keep yourself safe. Stripper can burn your skin and be very harmful to eyes so make sure that the gear includes good rubber gloves and face shield.

Sanding is another method to remove old paint - however if the paint on theses doors is very very old there is a possibility that it is lead bearing paint. If this is the case it is not recommended that you sand the paint at all and consider other alternatives to provide for containment of the lead based paint.

Spray and roll, more commonly called spray and backroll around me is when you spray the paint on and then another person follows behind you with a roller to roll the freshly spread paint. One of the reasons to do it is to get more even coverage in textured ceilings.

It's easier to do with two people so if it's just you might just want to skip the sprayer to keep things simple.

If you've never used spray equipment before it can take some time to get used to. Chances are you're going to mess up the first go (drips, sags, missed spots, etc) so at least practice on some cardboard or something.

I own and have used sprayers but I also own a Wagner Power Roller. It's a system that feeds paint through a tube into a roller so I don't have to keep a tray of paint. I eventually got good with the sprayers but with the power roller there was very little learning curve, no need to tape as much and cover everything. In almost every way I found it easier than the sprayer.

They make roller arms that attach to airless sprayers that do the same thing. There called pressure rollers. If you have access to the airless sprayer maybe you can find one to borrow or rent.