"Don't worry" here sounds like "Zan worry". Which accent is it? Is it common to pronounce it like this, or is it exceptional case?

Rebecca

Some particular accents from the UK will cause the long "o" sound like an "a" when pronouncing certain words. This may be why you heard "don't" as "dan." You don't need to worry about imitating this accent, as it is not common outside of the UK.