I don't know if this will help - it's something I use and it's worth a try.
Take a simple major scale. C major will do. (Tamsin - non-piano-players can do it on a flute or whatever, but if you can get hold of a keyboard - any keyboard - doesn't much matter for this excercise as long as it's in tune

it will give you a different visual picture of the whole thing. You don't need keyboard skills either - just play the white notes from C to C; C is left of two black notes).
Play the scale through normally to fix the tonality in your mind. Slowly - really listen to each note.
Then play the same scale, but instead of finishing on C, finish half-way up or down by playing F# G. eg CDEFGABCBAGF#G.
You'll need to do this a number of times, and practise approaching the ending from different directions, eg CDEF#G; CBAGF#G etc.
You need to be able to hear when that F is F and when it's F#.
That's the modulation to the dominant. If you can spot it when it's in the melody, it's the first step to hearing it when it's in the harmony.
Once this is starting to be more familiar try the subdominant - B becomes B flat and the melody will end on F. Practise listening to the difference between CBAGF and CBflatAGF.
I hope that's reasonably clear. It's easier to do than to describe.
It may sound terribly basic, but it sometimes helps in hearing the direction a piece of music is going. Try to listen to as much music as you can too - particularly pieces using conventional harmony; short classical pieces are useful.
Listen for the sharpened fourth and flattened seventh that you played in the scale exercises.
good luck
Maggie