Did Spock ever struggle with his humanity on the original series?
Kind of depends what you mean. Did we ever see Spock struggle with his humanity
as such, i.e. struggle with the fact that he was half-human? Narrowly defined, not that much. Journey to Babel, certainly, and TAS's Yesteryear. But even there, Spock was not ashamed of being half-human. Rather, he seems to have suffered as a result of others' tormenting of him as half-human as a child on Vulcan, and (if "Miss Jane Wyatt" is to be believed) overcompensated as a result.
Now, did we ever see Spock struggle with control of his emotions which, as a Vulcan, he must more strictly control than would a human? Yes, several times, in some of the most powerful moments of the series (most obviously the Naked Time, This Side of Paradise, Amok Time, All Our Yesterdays, and the Tholian Web, but also in the highly underrated post-depersonalization scene in Plato's Stepchildren). Being half human, this struggle presumably would have been more difficult for Spock than for other Vulcans. But it nevertheless would have been a struggle for any Vulcan; they have incredibly strong emotions, so strong they almost destroyed their own civilization.
This goes to a very interesting point that only sank in with me when I re-watched the series a few years ago after having not watched it for several years. Spock's half-humanity really is not part of his character in the narrative sense, and he is almost always treated as full Vulcan by the writers-- his actions and his habits of mind those of a Vulcan (often juxtaposed with those of the humans around him), his interactions with bridge crew those of a Vulcan to a human, even his physiology (which crops up as either complication or deus ex machina more than we'd like) that of a full Vulcan. (McCoy never stops himself mid-complaint about Spock's organs to say-- oh, great, there's your gall bladder, just like it would be for other humans). Narratively, his humanity is really only relevant to Journey to Babel, which admittedly is an absolute banger of an episode.
Where Spock's half-human identity
is relevant, however, is in a macro sense, as a sort of shorthand cue to the viewer for irreconcilable inner conflict and tumult. Again, strictly speaking the tumult/conflict/emotional struggle would be there for any Vulcan (that's the entire basis of their civilization) but Spock's half-human identity allows the viewer to feel this struggle more viscerally. Maybe this was all accidental (I'm guessing they originally made him half-human because they worried a full Vulcan might be indeed too 'alien' a character for the audience otherwise) but together with Nimoy's performance led to something (the deeply conflicted individual who nevertheless manages to be highly effective and the coolest guy in the room, to boot) to which many people were immediately drawn