Don't worry for a second about staying in either the Tenerife or Costa Adeje. Both are beautiful and as they are part of the same complex, I don't think it really matters which you stay in. The main difference I noticed was the Tenerife side seemed quieter, owing to most of the pool activities/daytime entertainment being over at the Costa Adeje, as they have the larger pool area.
We stayed in the Costa Adeje last year and have rebooked for this August. The brochure (First Choice I think) has it wrong in my opinion when it labels the Costa Adeje as 'adult focused' and the Tenerife as 'family focused'. If anything I would say the other way around. Both hotels are suitable for couples and families, but not for singles, unless they are looking for a quiet get away from it all holiday.