Using another anaolgy, imagine "i" is a photon and "." is a period of time between one photon and the next:
Moving away from the light source photons would be i...i...i...i...i...i...i
and moving towards the light source would be i.i.i.i.i.i.i.i.i
where if you were stationary in relation to the light source it would be i..i..i..i..i..i..i
Would that be an accurate anaolgy?
With time dilation always occuring, no matter your movments either towards or away from the source, dosn't that make "time" remain constant with respect to the speed of light?
If the speed of light never changes, how can the time interval between photons ever change? It shouldn't be different from i..i..i..i..i..i..i because they'll always move at c, therefore the 'next' photon will be emmited by the source at c after the standard ".." period - which would reach you at c after the ".." amount of time?
I hope what I'm trying to say makes sense. I hate not being able to convey an idea very well.