As Lauren Ipsum's answer states one should be wary of repetitive use of words, phrasings, and sentence structures. Such can easily be seen as poor writing. (Note that this effect might be intentional. For example, repetition in a character's speech or thought could indicate a lack of coherence or clarity, perhaps from distraction, fatigue, idiocy, anxiety, or intoxication.)
However, one must also be careful to avoid replacing an invisible repetition with a discordant alternative. Simple, common words (and sentence structures) often pass unnoticed where a less common form of expression would draw attention to itself. The connotations, tone, or rhythm of a replacement may conflict with the intended mood (e.g., "however" has a more rational connotation drawing the reader away from a sense of emotional conflict in your example text). Such replacements can also give an impression of purple prose (e.g., see this answer referencing the "said" bookism).
Intentional repetition can also be used to link items, to lull the reader, to provide a structural framework to the text, or to establish an expectation.
The brown brick building had seen better days. The rain was letting up, but as a car coughed past, it splashed a casual insult of brown mud onto the stairs to the apartment. A man in a worn trenchcoat pointlessly pressed down his soaked brown hat, his weathered brown boots trudging up the stairs.
While that might not be great prose, one can see how the repetition of "brown" both adds a sense of weary dullness and links the aged building, the mud (the casual insult), and the man who is doubly "brown", from top to toe. One could imagine a short story with such a beginning (and further less dense repetitions of "brown") might end with a twist on the expected meaning of "brown" like:
The estate lawyer's assistant looked dubiously at the boy who had been playing in the mud. "I was told that someone in this neighborhood could tell me how to find a Mr. Douglas Brown."
Obviously repetition must be handled carefully, especially when used to lull the reader. A dull paragraph might effectively hide a clue in a mystery novel, but a dull chapter would tend to cause the reader to abandon the book.
Consider the effect of the repetition of "but" with a few relatively small changes to your example.
My heart racing, I wriggled to the edge of the bed and fumbled for Travis' hand. Just make sure to hold onto it, he'd said, whenever you get the chance.
I searched for a few minutes until I finally found it—smooth and warm, sprawled on the cold surface of the floor. I gave it a light squeeze, but there was no response.
Maybe he was asleep after all. Or just pretending to be? The easiest way to find out was to get up and ask him. Yes, that was what I had to do; but I didn't.
Funny, I had no problem at all with the idea of killing myself, but I was a complete coward when it came to love.
While using short paragraphs increases the pacing (as appropriate for the character's anxiety), the repetition of "but" for the last clause of three paragraphs intensifies the emotion, almost like a hammer on an anvil: ding, isolation, ding, fear, ding, self-accusation and frustration. The repetition of "but" also intensifies the sense of the character's conflict ("I want, but ... I want, but ... I want, but").
By structuring the paragraphs in this way, there is little ambiguity that the repetition is intentional.
(Side suggestion: You might consider using "My hand searched for a few minutes until it finally found his, smooth and warm". Having the character's hand searching intensifies the focus on the physical contact; replacing the dash with a comma softens the contact, expressing comfort rather than shock in the finding and in the tactile pleasure, perhaps also slightly intensifying the "but" clause by avoiding the earlier abruptness of a dash.)