Definitely a place to avoid on Christmas Day. It's something of a myth that Australians spend Christmas Day at the beach: some might visit it, but most have their Christmas lunch at home, the same as in the rest of the world. Consequently the beaches are largely filled with international visitors trying to have a "typical" Australian Christmas...

In terms of driving to Wollongong, Christmas Eve will be busy, as will Boxing Day (26 December). The Monday and Tuesday (27 and 28 December) are also public holidays, so expect busy roads then too. The quietest day by far is likely to be Christmas Day itself, if you can do the road trip then. A sensible alternative might be to go by train (quite a good train service – it's only about 50 miles, after all): check the timetable for holiday arrangements.