I'm pretty sure you don't need an empty slot. Whenever you pick up an excess char, to recruit, sell, or save, they just kind of follow behind the rest of the party until you leave the dungeon.
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This is correct. you do not need an empty slot. Also, as a tip if you don't want to wait until the dungeon is lower level, just know that it will remain there even if you fail to complete the dungeon (had to retreat) and you can try again. Also rescuing them and then fleeing the dungeon still grants you the rescue and closes the dungeon, you just don't get the item the dungeon was offering.