Old Testament Parallels is a good place to start if you're interested in the context of biblical literature, as is Ancient Near East: Historical Sources in Translation edited by Mark Chavalas.

The Context of Scripture series is more comprehensive and generally has pretty reliable translations.

• The Context of Scripture, Volume 1: Canonical Compositions from the Biblical World

• The Context of Scripture, Volume 2: Monumental Inscriptions from the Biblical World

• The Context of Scripture, Volume 3: Archival Documents from the Biblical World

For Akkadian and Sumerian literature, respectively, the best sets of translations are Before the Muses: An Anthology of Akkadian Literature by Benjamin Foster and The Literature of Ancient Sumer by Jeremy Black et al. The latter is a print compilation of the translations available online through ETCSL.

You may also want to check the excellent editions of the major Mesopotamian mythological texts published in the State Archives of Assyria Cuneiform Texts series. In addition to translations, the books provide the cuneiform and transliteration.

Most of the Ugaritic texts are not readily available in English translation, but Stories from Ancient Canaan by Michael Coogan and Mark Smith is a great place to start for Ugaritic mythology. Pardee's Ritual and Cult at Ugarit and Parker's Ugaritic Narrative Poetry are useful as well.

I recently listened to Irving Finkel's The Ark Before Noah. (a really engaging book about the various cuneiform tablets that exist with the Ark story, including one he translated that puts a whole different perspective on what the ark looked like as well as a really interesting explanation of how he feels the Ark story developed through time and place into Jewish and then Christian (and Muslim) canon). He's an assistant keeper at the British Museum for Ancient Mesopotamia (including Assyria). This is his page on their website. https://www.britishmuseum.org/about_us/departments/staff/middle_east/irving_finkel.aspx If noone comes up with any other ideas, I suggest you drop the department an email asking them what they'd recommend. I doubt they'll mind as the British Museum book shop website is currently not functioning so it's not as though you can check there first at the moment. [edit the BM website says it's not functioning, but there is a website that functions for the BM online shopping as a whole. So I may have misled you there, but it doesn't look like the wide range of books that the BM used to have a decade ago when I was last in London shows up on the website.]

I've got Ancient Near East - An anthology of texts and pictures by James B Pritchard (editor) on my wishlist of books I'd like to read, second hand copies are cheap in the UK. But it was published initially in the 1960s and I have no expertise in the area so I'm afraid I can't tell you if the translations in it are good nor if it covers the particular texts you seek, nor whether or not it's been updated with any new knowledge since.Here is a link to the third edition, you may be able to work out if what you need is there from the Index? https://wordandsilence.files.wordpress.com/2017/01/ancient-near-eastern-texts-relating-to-the-old-testament-i.pdf