I’d pretty much second Chas here, with the proviso that, on the majority of (but NOT all) 8-key flutes and also on most keyless modern ITM flutes I have had the chance to experiment with, the Cnat at the top of the 1st octave is usually best (i.e. strongest and most in tune) fingered oxo xxx(+Eb - if you have it), rather than the commonest ITM fingering of oxx ooo. Equally, on most whistles (and cylindrical poly-flutes like Tipples and Dixons) oxx xox is usually best.
I mostly use oxo xxx on flute and oxx xox on whistle and, with long familiarity, mostly have no trouble switching between them according to which instrument I’m holding. However, I have a piccolo (acquired last summer) that requires oxx ooo, which I am less used to and at present have to do specific practice for if I’m going to use the instrument out. Because I mostly play keyed flutes, I also use the C nat key for the 1st octave C, switching between key and cross-fingering according to which offers the greater facility in a specific musical context. Odd and wonderful thing, the human brain and ingrained habit! Yes, you do have to work at it to begin with, but you get there in the end and, as with all these early stages issues, in the far future you’ll hardly recall you had the problem, unless you start teaching!
I’d add that both these fingerings make “over the break” movement easier than the oxx ooo fingering - less fingers moving simultaneously - and even make true Cnat rolls and D rolls with C nat as the lower note possible.
For the Cs nat and # at the top of the 2nd octave, again, I’d agree with Chas, though (and yes, I realise we are addressing the proud owner of a brand new keyless flute) he doesn’t mention the need to vent the Eb key for most of those fingerings on a keyed flute, nor the basic 8-key top C nat fingering which is xox xxFnato Eb. Also oxo xxo sometimes works tolerably in both octaves on some flutes, especially in swift alternation with E for the lower one.
The main point to absorb is Crookedtunes’ one -
You should look at all the options and figure out what works best for you, and on your particular flute
And you wrote,
I don’t know, should I maintain my poor-sounding fingering or learn the new one? It’s so hard to learn things new again after you’ve learned them once
My answer is unequivocally “Learn what works best” - you’ll thank yourself later that you made the effort and it will increase your adaptability and general command of the instrument.