I (U.S.A. citizen) currently have a Dutch residence permit for purpose of family reunion. When my sponsoring family member will leave the Netherlands later this year, I will also have to turn in my residence permit.

At that point my goal is to switch over to a Schengen tourist for up to 90 more days.  The Dutch immigration office told me this is ok but I'll need to exit then re-enter the Schengen Area, and she gave the example of taking a one-day trip to the U.K.

Instead of the U.K. I've wanted to go to either Andorra or the Faroe Islands.  Neither of these are Schengen, but my understanding is that due to their locations they do not have border controls with the Schengen zone.  Is this true?

Would that further imply a weekend trip to the Faroe Islands or to Andorra might *not* work to switch my status from long-stay resident to short-stay tourist (i.e. no place to get a new passport stamp)?