I would go first with the replace the doors option. That to me would be the best option, and shouldn't be to bad cost wise.
Otherwise all are viable options, just depends on the tools available and your inclinations. Best IMO would be to clear the paint, adhesive (I like Mike am a liquid nails or similar fan for this type of project), and brad or clamp. If the paints in good shape and you aren't wanting to repaint the entire face, you could just edge sand an inch or so down to bare wood. That'll give you bare wood contact for the adhesive and the transition from bare wood to paint will be hidden under the stop.
If you don't want to go through the hassle of getting down to bare wood, which I can full appreciate, I'd still opt for the liquid nails but I'd drop the brads in favor of screws from the back to give a little more mechanical hold.
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