I know almost nothing about solfege, except that trying to do it annoys me. 🙂 My children - who already knew how to read music on the staff, both for piano and singing - had to learn a bit of solfege at their previous school (b/c the school was against children reading music on a staff before 5th grade), and they (and I) felt that it got in the way of actually singing. But I assume it must be useful for something, since some professionals use it.
For me, once I have the first note, I'm mostly just recognizing intervals; if they get to be too large (say, an octave+), I'll play the note after the jump to make sure that I'm singing the correct note. And unless I'm singing with someone who's struggling to find the note on the piano, I never say the letter name - it's just bum-da-da-dum-bum-ba-da.
I personally would discourage the habit of always trying to sing the letter or solfege name of a note before one has learned to sing/feel the intervals. Maybe this is idiosyncratic, but I feel that immediately tying a verbal name tag to each note or interval promotes an over-focus on each individual step, to the detriment of getting a quick overall feel for the melody. (Iternabe, I don't recall your profession, but for some reason I think you're an engineer, so maybe the following will make sense to you: If a kid is learning to write math proofs, you want them to have an overall sense of how they're going to approach the proof - the general arc, with some feeling of which intermediate points they're going to have to hit on their way through - before they start writing. Once they have a general idea in their head, then it's on to the nitty gritty of figuring out the exact details of how to get there. I find that kids who focus too much on trying to pin down the exact wording of what Step 1 of Lemma 1 should be before moving on tend to get very stuck, because they don't have a broader picture in their head to keep them from getting lost in the weeds. Trying to sing the melody first is something similar - almost surely I don't get it exactly right at first, but it helps give me a feel for the lay of the land, and then when I play the melody on the piano, I have a clearer picture, not least because my piano is definitely going to be more in-tune than my voice. 🙂 At least for me, it's an iterative process, and I've found that expecting too much perfection in each step right off the bat isn't helpful.)
