It's difficult for Excel to determine if a date is in a cell because Excel stores dates as numbers but the format of the cell is a date format of some kind. What's more, if someone enters a date like 24th Dec 23, Excel won't necessarily recognise it as a date even though it's obviously a date to us humans. It is possible but it's convoluted and still not a certainty that it'll get it right.
So to avoid all that, would it do if either the cell wasn't blank, or perhaps was a number in order to decide whether the conditional formatting showed or not? It would just be a matter of adding a 3rd argument to the AND part of the conditional formatting formula.