According the the Merriam Webster site, only can be an adjective...
or an adverb...
depending on the word modified. That means that the classification is going to depend on what the sentence means. I believe that in your examples, the sense of the sentence changes as the word "only" floats through from place to place. We would have to judge what each sentence means in order to decide the role of the word "only."
Only I like my cat.
I like only her.
The word is an adjective meaning only I (nobody else), or only her (nobody else).
SLIGHTLY AMBIGUOUS EXAMPLES
I like her only.
I only like my cat.
Does the first mean that I like her (but only like, not love, an adverbial sense), or that I like her (but only her, not him, an adjectival sense)? Does the second mean that I like the cat (but only like, an adverbial sense), or that I like the cat (but nobody or nothing else, an adjectival sense)?
I'm not sure I see any consequence to classifying the word "only" as an adjective or otherwise, but I think that's a fairly good analysis.
P.S. Merriam-Webster uses the words "only one left" to illustrate the adjectival use, and "lost only one" to illustrate the adverbial use. I don't see how they're looking at it. Maybe they understand Orwell to mean, absurdly, that he only lost one, he did not tie it or win it. It seems more natural to me to think that Orwell meant that he lost only one, not more than one; that would lead me to classify the example as adjectival, despite the word order.