I honestly do not see the need to use SS for the handles, but it's your cooker, so go for it if that's what you want.
Yes, the wood screws that are used do have a sharp tip. Are you looking to replace both wood screws and the screws for the plastic / nylon handles ??
I have found wood screws at my local Lowe's that were the same size, albeit, they were Robertson (square) head, not Phillips. I think they are zinc coated.
As far as replacing the other screws, those might be tougher to find.
You might want to check your local Home Hardware store. I find that they carry stuff that the big box stores do not. Good luck!