I often work with translating documents into English. My translation style is more say-it-as-it-is than anything else. This is fine for news and similar-type material, but when it comes to literature and the like it doesn't really yield expected results.
How can I make my translations more "literary" or more descriptive in a story-like manner?
There is no such thing as a single "literary" style you can imitate.
It is crucial to understand that every text is created with a purpose. This aim is more evident in case of non-fiction, where we can usually clearly state: "This article was written with the purpose of informing the reader about X or Y," or "This advertisement was written with the purpose of convincing the reader to but Z".[1]
But there is also an aim to every piece of fiction. It's easy to think that the only purpose of a piece of fiction is to entertain, bring aesthetic pleasure or evoke emotion in the reader, but that is by far not the only possible aim. When writing a memoir or historical novel, the author's aim is to inform you of a specific story. When the novelist presents a character from an ethnic minority who speaks in a dialect, their purpose may be to throw light on this group's culture. When someone creates a character with specific, clear ideals, they may be intending to persuade the reader that these ideals are true.[2]
And language is the tool for accomplishing these aims. This means that as a translator of fiction you must first of all identify the aim of the style used. Just to give you an example:
Is the language slightly archaic? Why? Is the author trying to imitate a specific time period? Or are they only trying to evoke nostalgic feelings?
Is the language very metaphorical? Why? Is the narrator / point-of-view character a person with a lyrical temperament? What does the style tell us about this character?
Does the language focus strongly on some aspect of the world (to use the most cliched example, detailed and emphatic descriptions of the seasons)? What does this tell you about the world that the author is trying to present?
Only once you have identified the "subtype" of the literary language and you know what purpose it plays, can you look for an equivalent style in English. If you do not have a natural feel for the style, look up works of a similar genre and observe the literary devices they use.
Most importantly, resist the urge of applying some sort of unified "literary" English in your translations. You may just as well come across a text written in a say-it-as-it-is style, where your natural tendencies will make your work easier.
Side note: your question opens a few cans of worms related to translation theory. It would take at least a decent-sized paper to describe them in detail, so I'll just signal the problems:
One thing to have in mind is that different languages have different literary traditions for different genres. What is common for a memoir in country A may be a very unusual style in country B. The translator must choose whether to adapt the conventions to the target culture or not.
While ideally an analysis like this should be done with every aspect of every text translated, realistically (especially in market reality), it is not viable. Depending on the way you work and the time you have, doing even a surface analysis will help you home in on the proper style.
[1] Here's a summary of Katharina Reiss's "text types", one of the most commonsensical pieces of translation theory on this topic: https://www.slideshare.net/nobedi12/katharina-reiss
[2] Strictly speaking the translator should be concerned with what's in the text, not what the author intended, as these two things are often not the same, but I find it's easier to think of the writer's intentions.
