Yes, "will approach" would have been wrong. Following 'as' or 'when', you need a present tense, not a future. "As the date approaches" is grammatically correct, but not very natural. The phrase you need may not involve a verb at all - possibly the best way to say this could be the expression 'nearer the time'. For example, if there is a key date at the end of March and you are going to give the person information about this date during the third week of March, you could say that you'll give them this information 'nearer the time.'

As the date is drawing near,

The expression is "the day approaches." Can you please provide the sentence so we can see if this phrase works in context? Thanks.