That is, if X is logical, is it necessarily rational? What's an example of something that is rational, but not logical. If logical does not necessarily imply rational, what's an example of something that is logical, but not rational?
It depends on the understanding of rational. The question scope is too broad, so it makes an ill-formed question for this forum.
In general, something rational is just related to reason. And reason is not always logical. For example, a lot of people make rational decisions (that is, they develop a set of reasons) which are not logical and lead to poverty. There, clearly, reason might contradict logic.
In formal disciplines, there are multiple understandings of the term rational. Often, it is directly associated with logic. Usually, the term Rational in a technical document describes the logic of the content. There, logic and rational are the same. But it depends on the discipline. In psychology, it could be the antonym of unconscious.
In philosophy, there are also multiple uses. Rational, for example, can be the opposite to empirical. In such case, comparing logical to rational would be comparing peaches with oranges.
In the same way, a rectangle can be or not as a square. A rectangle is as a square because both are geometric forms, but a rectangle IS NOT as a square because both have different forms.
I'd say that rational to logic is like draftsman's skills are to a setsquare with which he is able to draw a square.
Rationality means acting according to reason, so a rational person would be a person, that often acts according to her reason.
Now, your confusion is driven, I think, by the fact that classical logic is the most often used "tool of reason", if you will.
I can totally not use classical logic during my act, maybe I use some other logic or not a logic at all, but as long as my act is performed with the use of reason, and not some other faculty, this act should be called rational and not (classicaly) logical.
I suggest that both are poor approximations to truth. Rational suggests that a ratio exists, for example the statement that one person in ten is left-handed. Logical is typically used in the context of binary decisions, such as if you are old and you are a man then you are an old man. So one is typically used in the context of real numbers and the other in the context of integers or natural numbers. One might argue that natural numbers are a subset of real numbers, but I think that they behave quite differently and so it’s better to say that rational and logical are seperate things.