A word like headwear or headgear is a hypernym for a word like cap or hat; that is, it usually refers to a range of items in a particular category. For example, if I went to the headwear section of a department store, I might be able to get any of these items: a cap, a hat, a visor, a beret, or a headscarf.
Headwear usually refers to items such as sun hats, rain hats, ball caps, snow caps, and stetsons. Headgear might be used when the item is more protective, such as a helmet.
I'm unlikely to say, "I'll be wearing some headwear tomorrow." Instead, I would say, "I'll be wearing a cap tomorrow," or, "I'll be wearing a hat tomorrow." (Whether I use hat or cap depends on what I plan to wear. Some headwear is almost always a "hat", and some is usually a "cap" instead.)
That said, if I was organizing a golf tournament, I might say, "It's supposed to be hot and sunny tomorrow; bring some headgear."
That's a convenient way to let everyone know they should bring some type of protective headwear – although what each golfer chooses to wear might vary quite a bit:
As for the decline in usage of cap that you speak of, I don't think it's as drastic as you might think. There's a famous line in an old Christmas poem:
And mamma in her ’kerchief, and I in my cap,
Had just settled our brains for a long winter’s nap
Mama's kerchief and the narrator's cap are usually imagined to like something like this:
I agree; that sense of the word cap is in decline – partly because fewer people wear sleeping caps nowadays. But if you are talking about a graduation cap (i.e. a mortarboard), that is still commonly referred to as a cap, and that's the word I would expect to read in your announcement:
Going to party wearing my graduation cap.
As for "with" or "wear", you could use either one:
Going to party with my graduation cap.
but I think "wear" is the better choice. We go to parties with our friends, but we go to parties wearing headwear.