I agree that the fee is ridiculous and unless your going to need several pairs of shoes and bulky beauty product equipment (hairdryer/curling iron) you should be able to make a carry on work.
You are allowed a carry on suitcase/bag in addition to your purse. What I do is bring the BIGGEST purse possible and pack all my beauty products/toiletries in that. Then I bring a small suitcase (check the website for exact measurements, but I believe it has to be under 45 inchs combined. The suitcase I stuff with clothes (using a plastic bag to compress the air).
Airport security requires you to put any liquid products in a regular size zip lock baggie and display the baggie when going through security. no products can be over 3/4 oz. and they will give you hell if they find you concealed a bigger bottle and/or didn't put it in a baggie. They sell airplane travel sets to put your liquids in at drug stores, that way if your bottles are too big, you'll have small enough bottles.
Remember that you'll need to take off your shoes when going through security, so make sure you wear matching socks (my mistake...lol).