From what I've experienced and learned from other sight-singers, we generally employ two primary strategies for singing melodies. One is the awareness of the scale degrees in relation to a tonal center, and the other is knowing how to sing each interval and singing the melody as a string of intervals.
Each strategy has its advantages in different contexts, and ideally we learn to use both strategies and how to switch between the two from moment to moment, as well as synthesize the two strategies into a complete skill of sight singing.
There is a third fundamental skill that is useful for sight singing, and that is to be able to sing the common chromatic alterations of a key along with being able to sing the diatonic scale degrees of the key.
It seems to me that you might be primarily leaning on the strategy of knowing how each scale degree sounds and also not yet skilled in switching between strategies.
As usual when we plateau in learning a musical skill, one great path to breaking through is to return to fundamentals:
- Learn to read and audiate all melodic intervals of less than an octave and use that skill to approach chromatic notes based on the interval with the previous note.
- Study and sing chromatic alterations of scales and modes.
- Study and sing the chromatic scale and the whole tone scale.
- Learn to audiate the most common chromatic notes in the context of the key/tonal center.
For the first bullet point, if you're not sure how to effectively learn melodic intervals, I suggest asking a separate question on this site and I can go into details in an answer to that question. Even a summary here would be too long for this answer.
For the second bullet point, I suggest making part of your daily exercises a singing of the major scale pattern starting from each degree. In other words, sing do to do an octave higher and then back down, then re to re, then mi to mi, etc. Octave shift if you have to. Next level of study is to follow the same pattern with the minor scale. Then sing the same pattern with the harmonic minor scale (singing ti instead of te or si instead of sol in la based minor). Finally, sing the same pattern using the melodic minor scale. Details on these varieties of minor scale would also be a subject for a separate question if you're not already familiar with them.
The chromatic and whole tone scales can be very challenging, especially in terms of arriving at the next do with proper intonation (it's easy to get quite flat singing up the chromatic scale). The study and practice of these will help unlock your sense of these notes in relation to the key center as well as your ability to approach a chromatic note as a half-step away from a diatonic note.
Finally, in terms of the "most common" chromatic notes, the melodic and harmonic minor scales will get you started on this. In a major key, we would expect to see te instead of ti and fi instead of fa more often than other alterations, since they are part of modulation to adjacent keys. In minor keys we are very likely to see ti instead of te and la instead of le as well as sometimes mi instead of me and even fi. Beyond that we have di occasionally as a leading tone to a ii or II chord. The study of the music theory of secondary dominants, AKA applied chords, will help in understanding when and why we will see these alterations.
In summary, we unfortunately often cannot efficiently acquire certain skills by merely "hacking away" at them as we encounter them. Rather, we often have to return to building primary skills that will help us overcome the difficulties in the main topic we are trying to learn. This is definitely true with sight singing and being able to modulate with confidence.
