'Наш отпуск в Нью Орлеансе был бы лучше, если бы не Щугор болл.'

'Our holiday in New Orleans would have been better if it hadn't been for the Sugar Bowl.'
'But for the Sugar Bowl, our holiday in New Orleans would [have] be[en] better.'

(It would help if I knew what a Sugar Bowl was.)

'Если бы не моя мама, мы все еще жили бы в старом доме.'

'But for my mother, we would still be living in our/the old house.'
'If it weren't for my mother, we would still be living in our/the old house'.

Perhaps I'm misunderstanding the Russian entirely, but I'm pretty sure that 'если бы не' is close in meaning to 'but for'. My translations may be wrong, but they're grammatical .