Hi Mayla! I'm not a voice teacher but there are several on the forums who will no doubt be able to help you from their perspective.
I'm a Kodaly specialist so I do pretty much ALL of my teaching without an instrument. WHAT AM I SAYING??? I do my teaching using the best instrument that we all possess - the voice!!
It is interesting that you say 'It just seems a little mysterious to me as one does not have an instrument sitting in front of them to place the hands/fingers on and establish some kind of external sensation for moving from note to note in music.' I actually think the opposite! It is impossible to create a sound with one's voice that has not been imagined first. The voice is an internal instrument and therefore anything learned through singing goes far deeper than something learned using an external instrument. It is the internal sensation that is more valuable!!
It is perfectly possible to learn to play an instrument to quite a high level without ever understanding the essence of music - anyone intelligent and well-co-ordinated can learn to do this - and you don't HAVE to have imagined the sound first before you play it (although brass and string players in particular have to do this more than pianists).
Singing is, in my experience, the key to understanding - and solfa is a wonderful tool to use in this regard.
Hmmm - yes, it would be boring to just plod through solfa exercises - I think my best advice would be to relate every solfa interval that your students learn to 'real' music. Kodaly teachers usually introduce one pitch at a time, and in a particular order, for various reasons - starting with pentatonic melodies as this is an excellent strong foundation for diatonic work. Having said that, however, the ears of UK residents (are you in the UK?) are very used to the sound of diatonic music as it is strongly in our folk heritage as well as art music. So if you are starting adults off you might use diatonic music rather earlier than we would normally do. However, I would still start with pentatonic melodies for children.
Using handsigns (developed by John Curwen in the 19th century then adapted by Zoltan Kodaly in the 20th) are a great help with 'cementing' the sound of intervals in the ear and the voice, as they provide a kinaesthetic link. They are also immensely helpful for sight-singing (the student sings the melody that your handsigns show).
If you would like to know more about the Kodaly approach, please just say - either on this board or in a Private Message. If you did some training I think it would open up all sorts of possibilities that would help you teach your students to read and to understand music alongside learning repertoire and technique.
Good luck and have fun!