"when our servers are under high traffic pressure" - this is a bit awkward I agree, but only the last three words.
If we rearrange it to "when our servers are under pressure from high traffic", I think it sounds good. It's using a metaphor, and I think that should be encouraged. It's interesting. And the phrase "high traffic" conveys some drama.
"your requests may take some time to receive a response from the server" - I think that's fine, to be honest. I like it.
I think you are conflating "awkwardness" with linguistic flair. Technical documentation English has become standardised to a large degree, which of course is useful, and efficient. But it is also a narrow usage of English, and breaking out of its straitjacket does not make language awkward.