You are on the right track with the idea of 'must'. The construction 'to be + to + infinitive' is used when someone is expected, scheduled or required to do something. It's a relatively formal construction, but it isn't old-fashioned. We often use it as a form of the future. For example, we can say 'The president is to begin his tour tomorrow', meaning that it is part of a formal schedule. So the expression 'you were to check' means that this person was expected to check, or that this was part of the person's duties. The second example means something like 'if we are going to have any chance of securing'. It's slightly different from the example above, and is a way of expressing a hypothetical situation. I suspect that if you were to translate this idea into Italian it would be a subjunctive.

1a) "you were to check" - The person speaking is saying that they told you previously to check (examine) all the slave boys. A more common way of saying this is "you were supposed to check" 1b) "you were to do" - Similar to 1a. The person speaking is saying that they told you to do (perform) a task. Another common way of saying this phrase: "you were supposed to do" 2) if "we're to secure" - Another way of saying "if we are going to secure." Meaning to safely obtain in this context. The way the text reads is not incorrect, but there are more common ways of phrasing these sentences in conversation.