General rule #1 - don't wear shorts with short coloured socks. It is fine for plain white socks, preferably covering the ankle. Most pro shops will sell you a pair if yours don't comply. No such issues with long pants.
General rule #2 - don't leave your shirt tucked out.
General rule #3 - don't wear a hat inside the lounge.
Occasional rule #4 - carts require a medical certificate. Not all courses you are visiting Michael, but some of them. If the course is mildly undulating (ie easily walkable) then don't presume you can take a cart. At some courses, if you take a cart (with a medical certificate) then you will probably also have to pay for a driver. I think Barnbougle is like that.
Golden rule - don't be a Dick.
The dress inside the clubhouse can (confusingly) be more relaxed than on course, eg boat shoes with no socks may be ok. I emphasise may.
Your last day at Royal Adelaide will be a walking day unless you have a medical certificate. You will enjoy the walk. Buggies are provided (push/pull buggies). You can hire battery-powered buggies at many of the clubs if you don't want to push/pull your own.
If you are taking shorts for golf, just pack a few pairs of white short socks. Make sure your shirts can be tucked in and have a collar. Bring a couple of hats (bucket hats are better than baseball caps). Sunscreen is available at most clubs. You will need it. UV today in Adelaide is 14 - you will burn much more rapidly in OZ than in USA. Burnt ears aren't fun (and worse if you eventually get a melanoma on one of them).
I think the days of coat and tie have gone. In fact you can (generally?) wear your golf attire including shoes after golf. If you were going to the dining room, it is probably best to have freshened up and changed but it might not be required.
James B
ps if you can, have a mix of bounce on the sand wedge and lob wedge. Some bunkers will be 'fluffy' suiting the bounce whilst other courses will have 'packed' sand which will just bounce any club with bounce. Some lies around greens can also be tight, so a wedge with bounce isn't always the preferred shot. Choose the right bounce club for the shot depending on the course/sand/lie.You are playing at so many courses in different states that you can't generalise about what you will find.