Possible confusion here over what you mean by “upper register”??? 2nd 8ve or 3rd?
Also, exactly which Dixon have you got? If it is the cheapest cylindrical polytube one, it will work best with different C nat fingerings than if it is the 3-piece polymer conoid R&R-derived one - and it won’t work terribly well at all (see/search old threads…), which could account for your problems towards the top of the 2nd 8ve..
Whichever model you have, if you’ve only had the flute a week and you’re new to fluting (?), it will take quite a while for you to get good control and the flexibility needed to adjust your embouchure optimally as you go up and down the range. Try exercises such as slurred octaves, &/or tunes with over-the-break jumps as well as long tones throughout the range…
There are plenty of old threads on C natural fingerings, but for that middle C nat (top of 1st 8ve) you’ll likely find oxx xox best (strongest, clearest tone and most in tune) if you have the PVC cylinder flute and oxo xxx best if it is the conoid polymer… but try both and all the other suggestions above on whichever flute it is, and decide for yourself.
If you meant the C nat at the top of the 2nd 8ve (which I doubt), then experiment to find which out of oxo ooo, oxo xxo, oxo xox, oxx xxo, xox xox and maybe some others works best for you. (Also, 2nd 8ve open C# will likely be way flat and oxx xoo will be better.)
It’s the cheapest cylindrical flute (one-piece) from Dixon, and I mean the 2nd register.
I tried OXO XXO last night and seemed fine, so ta muchly.
I’ve had a bit of a go with a Boehm flute (for a couple of months) and decided to try a simple keyless flute simply because I felt I could ‘feel’ my way into a tune or improvise more easily with the simple flute. This is my first instrument, so it’s pretty rough.
A version of ‘The Green Hills of Tyrol’ that I sang in a choir 25 years ago is stuck in my head, so I use that as a warm-up. Can only remember the first part (which runs from low D to 2nd register G), so the family is having to put up with rather a lot of repetition.