1 - Why are you overlaying the paneling, remove it, then assess whether the walls can be repaired or need to be overlaid.

2 - If you make the walls thicker than your door jambs, then you have to add spacer material to the jamb that will bring the jamb flush to the new surface. Let's assume that the jambs and trim are fine now and you overlay with 3/8" drywall, you will then need to add strips of wood to the jamb that are 3/8" thick as well, then install your trim over that.