You seem to be using the movable doh method, where doh is the root note of each key. This is more commonly called tonic sol-fa, solfege being the French name for the system which is used universally in France and several other countries, solfeggio in Italy - fixed doh, where doh is always C.
Whichever is used, this example is in C! You are familiar with the diatonic notes of C D E F G A B, singing or calling them as d r m f s l t (doh, ray, me, fah, soh, lah, ti).
Now, the changed notes - those for which accidentals in key C would be needed - are d, r, f, s, l, and could be sharpened; and r, m, s, l, t, which could be flattened. By changing the sound of each word/name slightly, it's not too difficult to sing the non-diatonic notes. Those sharpened are: de, re, fe, se, le. Can't find how these are pronounced, but 'der, rer' will suffice. Hope someone out there can enlighten! EDIT: Seems to be a short sounding 'e', so in C, G# will be said 'said' without the d sound.Those to be flattened will be changed with an 'aw' sound found in claw. They are raw, maw, law, taw, usually written ra, ma, la, ta. Obviously la and lah can get mixed up until one is used to the system!
Using this tonic sol-fah in, say, D, the F note gets sung as 'ma'; the flattened leading note (C) gets sung as 'ta'. In the solfege system, that C will always be labelled 'doh', whatever the key.
Another related thing is to practise singing not just major scales, but chromatics as well. And familiarise yourself with intervals - be able to sing a P4, a b5, a m7, etc. You may find that working through intervals will be a different slant on your sight singing.