I think you leave this form in present tense, then adjust the tense to past in whatever follows this in the sentence.

きのう、ごはんをたべないままでがっこうにいった ..

I used the 'submit new model' sentence feature to search some sentences with 'ないまま' (nothing with なかったまま came up), and a lot of them were in past tense.

That's right - い adjectives remain unchanged. You actually don't need a で after ながら (think of it as a verb ending), but otherwise, 帽子をかぶりながら is grammatically fine. I think there's a tendency to use ながら to express something that you are actively, intentionally doing at the same time as something else, whereas まま implies something left as-is, so かぶったまま sounds a bit more natural in this case, because the sense is that you left it on, i.e. neglected to remove it before eating. But if you check Google, you can find results for both cases (many of them, funnily enough, apparently about the acceptability of hats while eating! ;))