There are really several questions here:
Is the question of moral responsibility a valid one against determinism?
That depends what you think moral responsibility is based on and whether you think it exists. If you think a person can only be held morally responsible if that person has libertarian free will, and you think moral responsibility does exist, then you have to hold that determinism can't exist. The argument would be of this format:
- Moral responsibility can only exist if libertarian free will exists. (assumed)
- Moral responsibility does exist. (assumed)
- Therefore, libertarian free will exists. (by 1 & 2)
- If libertarian free will exists, then determinism is false. (definition)
- Therefore, determinism is false. (by 3 & 4)
However, not everyone agrees with the assumptions in 1 and 2. Compatibilists deny that moral responsibility requires libertarian free will. Moral non-realists (of many different stripes) deny that moral responsibility exists.
I feel like while everything is determined, our experience seems free. We FEEL like we could have done something other than what we did.
I'm not sure we do "feel" it as much as either a) live with the assumption or b) not think about this issue much at all. And unpacking exactly what we mean by "done something other than what we did" is not easy; Daniel Dennett devotes many pages to this in Freedom Evolves.
So couldn't you hold someone morally responsible for their actions if they thought there was a possibility of not doing what they did, but they chose to do what they did anyway?
Well, how does this consideration on the person's part change anything? If determinism is true, s/he still had no ability to do otherwise. In fact, s/he also had no ability to do anything other than pause and reflect about what s/he was about to do, consider the morally preferable alternative--and then do the less moral act anyway.