Did you unpair from the 5s? Gotta do that before you can pair to the 6s... unpair from the 5s... then backup the 5s to iTunes via USB cable and make sure you select the encrypted backup option to save health data from the watch... then restore the backup of the 5s to the 6s... then pair the watch to the 6s.
Might be having connectivity issues because the phone hasn't come out yet...I would imagine having the handset physically in hand would really help with any connection issues you might have with another Apple product. Good luck!