I don't like using chemicals either, but my gawd, it sounds like your cat has flea infection not just infestation. go get some ADVANTAGE right now, get it cleared up asap, and during that time you can research for a natural cure. i have heard that brewers yeast puts a smell fleas don't like in their skin but it didn't work for my cats, neither did garlic. ****go get some the advantage TODAY for right now because it sounds like the fleas have caused a skin infection and you should clear that up right NOW. don't squeeze them or bother the cat anymore than the poor thing is aleady irritated.
also, vacuum once per day. a really good vaccuuming.
i used to live by the beach and your situation sounded just like mine, then i moved inland and there were hardly any fleas, now i live in arizona and have 13 cats and i haven't seen even 1 flea (they don't live in dry hot weather). do you live in a humid or beach area?
either way, don't let the "chemical" thing keep you from helping your cat right now. get the infestation fixed today and the infection will stop. also wash her/his bedding with bleach and detergent and vaccuum like a fiend. put moth balls inside the vaccuum cleaner to kill the fleas you vaccuum up. the best vaccuum in the world for this is the ORECK. i've used them all and Oreck is superb.