What problems can one find with a utilitarian view of ethics?

One of the most important objections to utilitarianism is the objection from moral dilemmas. If one course of action can bring about more 'utility' or flourishing than another, then it is the moral thing to do - this is the utilitarian position. But let's consider a situation in which there is seemingly no 'right' answer as to what is the moral thing to do. For example, if I see a child drowning, and a doctor drowning, the utilitarian would have us believe that if rescuing the doctor brings about more lives saved, then saving the doctor is moral, and saving the child is immoral. What seems to lack from this account, is the residual 'badness' that would come from choosing one over another. Often in life, we are confronted with situations in which we have to choose "the lesser of two evils" - picking between two outcomes which are both bad. The problem with utilitarianism, is that it does not give us the tools to recognise this. A utilitarian would argue that there is no moral dilemma, and that if we want to figure out which person to save, all we need to do is evaluate the aggregate 'utility' their survival would entail, and leave it at that. This seems to be problematic since we do not want to easily say that saving the doctor is more moral than saving the child even if saving the doctor would save more lives. Rather, we want a system of ethics that allows for moral dilemmas, and includes a convincing account of situations in which there is seemingly no right course of action.
The other serious objection to utilitarian ethics, is that of moral luck. Let's imagine the following: two unrelated people drive their cars in a state of intoxication down a nighttime street. One of them manages to get home scot-free, the other ends up accidentally running over a child. The first driver was lucky, in that the roads were clear when they were going home, the second driver was less lucky, but are we inclined to say that the first one is more moral than the second? For a utilitarian, the answer would be yes. This is because utilitarianism is a consequentialist account of ethics - it only targets the consequences of actions as that which is 'moral'. But moral luck shows that consequences are not wholly within the moral agent's control, and that it is in fact the attitude or disposition of the agent that is evaluated morally. For instance, we condemn the drunk driver not for running someone over, but rather for getting behind the wheel drunk to begin with. Of course, it is the case that legally, the first driver is less accountable than the second one, but this seems to be more a case of luck, than the first driver acting in a more ethical way.