If you can tolerate some ales, why not start making them?
You can control the amount of carbonation through the amount of sugar you add at the final stage. Making beer is quite fun... and quite cheap (in Canada, anyways). I don't like much carbonation, so I only use half the sugar at the carbonation stage. You could probably cut it more; ask your local brewmaster. The carbonation is quite different to mass-produced fizzy drinks, anyhow... smaller bubbles that dissipate faster.
Hoping to get back into homebrewing in the UK; mostly illegal in Japan.
If you just want something flat and not too alcoholic, is sangria OK? Then you can buy cheap wine and flavour it with whatever fruit juice you like.
I have never met a cheap white I've liked. Reds, I like the more powerful acidic ones, so I'm not a good source of info. for you. I can't wait to try locally produced French reds!