Hold your horses, don't convert!
What size (corded length) circulars are you using? I noticed this was conveniently NOT mentioned in the pattern. Not a good thing for a newbie. If you're using something short or too long, this could be part of your problem.
The site's coming from Pennsylvania (where I happen to be coming from also). I'd say it's a US size measurement. Your ribbing is going to "give" later in the pattern. From the picture, it looks like a very nice headband is being created with the sizes indicated. Those 6 rows are the headband and the switch to a larger needle will be your "poofed out" part of the hat.
Since you have such a nice hat pattern there, I'd stay with the needle sizes they indicate and give it one, last, try before converting to another needle size or corded length.
Here's a few other random thoughts:
Show us a pic with the project on the working needle. It's a lot easier to see the problem than describe it, no?
It's also possible your ribbing rows are a bit tight (which made me wonder about the cord length) and it's making you think things are way off.
Check your tension to be sure it's not too tight nor too loose. You should see the loops on the circulars "give" a little as you slide your work along while knitting.
As you knit up onto those larger needles, the ribbing will make a lot more sense to you as your hat takes on it's shape. You'll find the conversion row a bit tough, especially if you've knit (way) too tight, like I usually do.
The hat's ending is pretty much as I described in my example, except the last 2 rows will decrease the diameter and shore up the top part of the hat, prior to the yarned circle routine she and I have described for you.
Remember, these hats only take a little while to make, so if you end up with something you'd rather not, it's not like the time you might invest in say, a sweater or coat.
Don't be afraid! Even if your first try doesn't fit you, you can always rip it back and try again, or donate it to someone who'll fit it!
Hang in there and Good Luck!