But ask me to sing with flats and sharps mixed in, or worse yet, sing something like: "1, 5, 3, 7, 2, 6, 8," and I get lost. I can sing the first note okay (if it is in fact, middle C), but I can't jump from that to E with any accuracy -- until I hear someone sing it, then I can sing it back.

Suggestion -- think of scales in terms of an orderly sequence of steps instead of a set of specific notes. If you can find the tonic and hear the difference between a half step and a whole step, you can figure out the scale. And once you have the scale figured out, you can start to think in terms of intervals instead of hitting a particular note in a scale. Each mode has it's own structure of steps and half steps.

For example, a C major scale is: CDE_FGAB_C
OR -- whole step, whole step, half-step, whole step, whole step, whole step, half-step

A D major scale is: DEF#_GABC#_D
same pattern!

A minor scale is:
whole step, half-step, whole step, whole step, half-step, whole step whole step

For example, A minor:
AB_CDE_FGA

Voila! If you know the structure, you can sing any scale. Now if you are singing in C Major, you can think of going from C to E as two whole steps, instead of trying to just know the right note when you see it on the page. Then the next note is just and interval and the one after that one... Or, "I come in a half step above that note right there..."

Anyway, it helps me relate a sound to the dots on the page.