John Huehnergard's A Grammar of Akkadian: https://brill.com/view/title/38184 seems to be well-respected. It's a grammar textbook that goes through the usual topics (nouns, verbs, etc.) Teaches the dialect of the Old Babylonian Empire and should be approachable for someone who has studied another language before.

A History of the Akkadian Language: https://brill.com/view/title/21744 was just published. It's monumental and eclectic. All sorts of topics related to Akkadian. Its internal history, changes, as well as recent history of its deciphering. I've been skimming that, using that as a sort of backgrounder. Some of it's over my head, and I took a few semesters of linguistics once upon a time.

The Chicago Assyrian Dictionary has been fully published and digitized: https://en.wikipedia.org/wiki/Chicago_Assyrian_Dictionary But it's like the OED and I think at the start you would want a wordbook or wordlist with your reading materials, that matches the dialect/era you're working with. A variety of those floating around.

The Oxford Handbook in Cuneiform Culture - more background reading in https://www.oxfordhandbooks.com/view/10.1093/oxfordhb/978019...

While the written form was pretty fossilized (like Latin or English spelling) it was used for ~3000 years and changed a great deal over time like any language. Old Babylonian seems to be a common starting point. I think probably because it's well-attested, was some of the first material discovered, and because it's the language of (the most complete version of) Gilgamesh, and also Hammurabi's law codes.